I have seen many debates on the existence of God and intelligence doesn't seem to be a deciding factor. I've seen both intelligent and ridiculous arguments coming from both quarters. Is it possible that some people have a sense of (presence)? Like a baby feeling it's mother looking over it. A feeling that their is a great force outside of ourselves. But like trying to explain color to a blind person they can be certain but lack a means of explaining why they are so certain.
Your question enters the field of psychology of religion.
We know that patients with psychosis hear voices and feel the presence of other persons who pursue them. In addition they speak to these imaginary people.
This observation shows that certain people feel the presence of other persons even when there are no other persons around.
These observations raise the question how to discriminate between psychosis and religion, the latter understood as communication with a personal, but invisible being, e.g. with the Jewish and Christian god, with Jesus Christ, with Maria, or with a series of Christian saints.
I consider the demarcation a difficult task.
Bertrand Russell discusses this question in Mysticism and Logic and A Free Man's Worship. Russell, being a hardened materialist himself, ultimately dismisses mystical feelings as illusions, but doesn't simply dismiss them offhand the way many atheists do, and he does consider them to be important aspects of our psyche and culture.
Wittgenstein was a student of Russell, and held many of Russell's positivist views, but was also a deeply religious person. He held that religious and mystical experience cannot be properly described, because it lay outside the boundaries of logic and language, but that did not in any way make it less real.
His famous quotes from the Tractatus:
4.1212 What can be shown, cannot be said.
6.44 It is not how things are in the world that is mystical, but that it exists.
6.45 To view the world sub specie aeterni is to view it as a whole--a limited whole. Feeling the world as a limited whole--it is this that is mystical.
Show his mystical side, despite his early philosophy being one of the central texts of empirical/positivist thinking that is usually the basis for denial of religious experience.
William James in his The Varieties of Religious Experience criticize the positivistic dismissal of religious experiences:
Vague impressions of something indefinable have no place in the rationalistic system.... Nevertheless, if we look on man's whole mental life as it exists ..., we have to confess that the part of it of which rationalism can give an account of is relatively superficial. It is the part that has the prestige undoubtedly, for it has the loquacity, it can challenge you for proofs, and chop logic, and put you down with words.... Your whole subconscious life, your impulses, your faiths, your needs, your divinations, have prepared the premises, of which your consciousness now feels the weight of the result; and something in you absolutely knows that that result must be truer than any logic-chopping rationalistic talk, however clever, that may contradict it.
Both William James and Bertrand Russell give some characteristics of what a religious/mystical experience is, in the above mentioned texts.
Aldous Huxley, himself an atheist, , puts forward in his The Perennial Philosophy that the different religious experiences are all aspects of one fundamental truth, that various cultures, prophets and mystics discover at different times:
the metaphysic that recognizes a divine Reality substantial to the world of things and lives and minds; the psychology that finds in the soul something similar to, or even identical to, divine Reality; the ethic that places man's final end in the knowledge of the immanent and transcendent Ground of all being; the thing is immemorial and universal. Rudiments of the perennial philosophy may be found among the traditional lore of primitive peoples in every region of the world, and in its fully developed forms it has a place in every one of the higher religions.
That mystical feelings, or Godsense as you describe it, exist cannot be denied. They cannot be dismissed the way the Richard Dawkins' and Sam Harris' of the world do, and any philosophy purporting to explain human experience without taking them into account is like a cook trying to describe recipes based on the chemical properties of food, while ignoring things like 'spiciness' and 'sweetness' etc...
The problem is in the leap that people make from the reality of these feelings, to the unjustified absolute certainty that many religious people have about their beliefs. Just because your feelings of awe and wonder are real doesn't make your belief in Southern Baptist Christianity (or Sunni Maliki Islam, or Progressive Scientology, <
insert your favorite tribalism here>....) justified, and it certainly doesn't allow you to treat everyone else who doesn't believe your own dogma like crap.
A whole new branch of research has opened up in the last few years called "neurotheology". Several research teams either have or are conducting research on religious experiences (i.e.; meditating) by scanning subjects brains in an MRI machine. There seems to be solid evidence of specific brain functions associated with such experiences. There is no agreement on what it means. Some claim it represents a "God" detector, others that it only means these folks are experiencing something - but it may only be a center for self-delusion.
It obviously is going to depend on who you ask.
For an empiricist, the answer to this can only depend on something we've detected with our senses. So, to confirm that there is such a sense, one of the other five senses would need to confirm it somehow.
Many, however, have argued that there is such a feeling, not so much by providing a philosophical argument but rather by their personal experience. For some-still-pretty-philosophical examples, see Thomas Merton, Holly Ordway, or CS Lewis.
In any case, Peter Kreeft has an counter-argument to the common "psychology of religion" argument: in general, one can only ask for psychological reasons for anything once one concludes that psychological reasons are appropriate. In the context of belief in God (or a feeling of God's presence, or whatever other similar aspect), that means that a blanket psychological answers beg the question - they assume the answer is that "God doesn't exist, it's all in one's head" and then look to explain why it is made up. (There are obviously examples of individuals that hear voices where psychosis is the reason.)
That feeling is the feeling that there is an overall Perceiver, and it is not from the outside, it is the innermost part of every person. Swami Vivekananda says on this (Complete Works, V7, p 54-55; also here under the heading Inspired Talks, sub-heading Wednesday July 17 - http://cwsv.belurmath.org/volume_7/vol_7_frame.htm):
Shankara further asks, can you see existence separate from everything else? Where is the differentiation between two objects? Not in sense-perception, else all would be one in it. We have to perceive in sequence. In getting knowledge of what a thing is, we get also something which it is not. The differentiae are in the memory and are got by comparison with what is stored there. Difference is not in the nature of a thing, it is in the brain. Homogeneous one is outside, differentiae are inside (in the mind); so the idea of "many" is the creation of the mind.
Differentiae become qualities when they are separate but joined in one object. We cannot say positively what differentiation is. All that we see and feel about things is pure and simple existence, "isness". All else is in us. Being is the only positive proof we have of anything…
Shankara says again, perception is the last proof of existence. It is self-effulgent and self-conscious, because to go beyond the senses we should still need perception. Perception is independent of the senses, of all instruments, unconditioned. There can be no perception without consciousness; perception has self-luminosity, which in a lesser degree is called consciousness. Not one act of perception can be unconscious; in fact, consciousness is the nature of perception. Existence and perception are one thing, not two things joined together. That which needs no cause is infinite; so, as perception is the last proof of itself, it is eternal. It is always subjective; perception itself is its own perceiver. Perception is not in the mind, but perception brings mind. It is absolute, the only knower, so perception is really the Atman. Perception itself perceives, but the Atman cannot be a knower, because a "knower" becomes such by the action of knowledge; but, Shankara says, "This Atman is not I", because the consciousness "I am" (Aham) is not in the Atman. We are but the reflections of that Atman; and Atman and Brahman are one.
So the feeling that the Perceiver is outside is a creation of your mind. The Perceiver is your innermost soul, or the Atman, and that innermost soul is one with Brahman.
Simone Weil from what I've read is commonly taken to have experienced a mystical event, though she did not consider to take herself to be a Christian in the orthodox sense; but this is prefigured in her own early writings at one of the grandes ecoles; Finch, wrote in his Intellect of Grace that
It is in one of these [early] writings, the Fragments on Feeedom, written in 1926 that comes closest. It shows how she understood thinking, the thinking that was continually born in her, and came to her as a gift and not as an effort of will. It was so closely connected to an absence of self and self-effort as to make the term 'thinking' and 'freedom' virtually interchangeable in her grammar. We must bear this in mind when we read 'This freedom is God'. The thinker here is not the Cartesian ego ('the self that knows the self') but the inspiring source itself, manifesting itself through us only when no ego or agent is in the way. (If we qualify it in intellectual or Christian terms, it is not far off to describe Simone Weil as Shamanic and charismatic).
He then quotes from this work, prefacing it with the comment he is 'speaking in intellectual tongues'
God is above all determinations, since he does not limit himself to a single idea: he is thus free, better expressed, he is freedom itself ... I can agree to call God my own freedom. This convention has the advantage of freeing from the God object...if God is my freedom, he is and exists every time that every time my freedom manifests itself on my ideas and movements, which is to say every time I think.
This is very different from the idea of 'a baby feeling its mother looking over it'; one might consider the extract as an intellectual idea of God, as what is taken to be her conversion from the agnostic athiesm she was brought up with (culturally her family were assimilated Jews) uses language that is very different: she speaks of 'conviction', 'compelled' and 'possession' - which is perhaps more aligned with your 'feeling that there is a great force outside of ourselves'.
Even though I consider myself a very devout Catholic and I firmly believe in God, I know I don't have a godsense. However, I also know there are people that believe they have a godsense. I think this is primarily due to some people wanting to believe that they have it, regardless of whether they do or don't.