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.
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.
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:
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:
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:
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, <
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):
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
He then quotes from this work, prefacing it with the comment he is 'speaking in intellectual tongues'
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'.