When can ZFC be said to have been "born"? The "History" section of the Wikipedia article on ZFC isn't particularly helpful. The only thing I have understood from it is that ZFC had appeared after 1922. In what book or paper was ZFC first explicitly formulated and proposed?

Reading in Kanamori's introduction to the Handbook of Set Theory, the text seems to support Robert Israel's answer.
On p.10 it says that von Neumann wrote in his Ph.D. dissertation an axiomatization of set theory, however it is not clear to me (from this text, and from other texts that I have seen this mentioned) whether or not his was the basis for the $\sf NBG$ set theory, rather than $\sf ZFC$ itself.
On pp.11-12 it says that in 1930 Zermelo published an axiomatization which was $\sf ZFC$ without Infinity and Choice (so perhaps $\sf ZF$ would be better here). So if we include those back into the axioms, it works out fine. There is no citation, but I suppose that sort of limits the possible papers.

The paper referred to by Kanamori is the following,

Zermelo, E., Über Grenzzahlen und Mengenbereiche, Fundamenta Mathematicae 16 (1930), no. 1, pp. 29--47.

And indeed on the second page of the paper it says "ZF". Kanamori indicates, as I wrote, that the axioms were missing both choice and infinity. However in his paper (which shares similarities to the introduction of the Handbook I was using before)

Akihiro Kanamori, The mathematical development of set theory from Cantor to Cohen, Bull. Symbolic Logic 2 (1996), no. 1, 1--71.

Kanamori points out that Zermelo assumed the axiom of choice as part of the logic. It is not fully clear to me whether this axiomatization was a first-order or second-order one, perhaps someone that can read German better than me can help with this part.
Once moving to first-order logic, the addition of choice and infinity into the list becomes inevitable, and we have the modern axioms of $\sf ZFC$.
