No, humanism does not, in any way, necessitate relativism. They are completely different philosophies. In fact, I'm not sure that I can even imagine a way in which humanism would imply relativism.
One thing to get straight at the outset is a definition of "humanism". There are at least two major ideologies that typically fall under the umbrella of humanism. First is secular humanism, which is probably the one you're thinking of when you talk about "humanism". Secular humanism tends to justify action in terms of human reason, ethics, and justice. And in doing so, it naturally denounces religious dogma, superstition, and other things that could be considered "pseudo-science". However, it does not prescribe a particular code or system of ethics, and many philosophers who fall into the "humanist" camp have strongly believed in universal moral standards, exactly the opposite of relativism. A few examples of such thinkers might be: Immanuel Kant, John Stuart Mill, and John Rawls. Certainly, secular humanism can be seen as incompatible with a strong religious faith, on which basis you may take objection to it, but it certainly is not incompatible with objectivism (with a little O), the idea that there is a universal system of ethics accessible to all.
Another common branch of humanism is religious humanism, which actually attempts to integrate humanism with religious ideals. Religious humanism is fairly straight-forward: it places the focus on the human being, affirming the individual dignity and worth of all people, just like the name naïvely implies. Certainly this could be compatible with religious teachings. Søren Kierkegaard is a famous "Christian Existential Humanist".
Beyond these two specific disciplines, you'll also see "humanism" applied generically to any philosophy that places the primary focus on human beings, as opposed to society at large or organized religion. But again, the idea that different cultures have different ideas about morality and ethical principles, and that those ideals are equally as valid as any other culture's ideals, is not implied by this line of thinking. Neither is the relativist tenant that there is no absolute truth or validity. In fact, a humanist could argue that human intuition, reason, and moral virtue are themselves absolute truths!
But beyond that, it appears that your confusion (and the confusion of whomever wrote the article you are quoting) stems mainly from the notion that atheism (or the rejection of organized religion) implies relativism, absent or beyond its humanist affiliations. That is also quite incorrect (albeit quite a common fallacy that religious scholars succumb to).
The flaw lies in thinking that the only possible source for an objective morality is from [a] God. While that's certainly one possible source, and even a good source, it's far from the only possible source. Humanists, as I hinted above, would argue that human rationality, intuition, and logic are the sources for morality, and since all humans share these faculties, such a moral framework would be objectively shared across all humanity.
There is an entire world of philosophers who don't necessarily believe in the existence of a supreme being, yet believe in objective morality. The one who makes this position most persuasively is probably Shelly Kagan, a contemporary moral philosopher. And other philosophers have even argued (this is Plato's famous Euthyphro dilemma) that objective morality cannot be based on God, even if both are granted to exist.