Short answer: it's probably Apollo.

Long answer: Socrates uses "God" or "the god" (either translation works) throughout the Apology to refer to the god who he believes chose him to become a philosopher, e.g.

God orders me to fulfill the philosopher's mission of searching into myself and other men

and

This is a duty which the god has imposed on me, as I am assured by oracles, visions, and in every sort of way in which the will of divine power was ever signified to anyone.

These are clear references back to this story:

I will refer you to a witness who is worthy of credit, and will tell you about my wisdom - whether I have any, and of what sort - and that witness shall be the god of Delphi. ... Chaerephon, as you know ... went to Delphi and ... asked the oracle to tell him whether there was anyone wiser than I was, and the Pythian prophetess answered that there was no man wiser. ... When I heard the answer, I said to myself, What can the god mean?...

The god of the Oracle at Delphi is Apollo.

This is from a non-peer-reviewed journal, but this article has some relevant info under 19a. I'll quote the relevant part here:

Socrates often refers to "the god" (ho theós). If Socrates is thinking of a particular god, and he is, he never does actually name the god, though later we will have no difficulty understanding who it is -- the "god at Delphi" is Apollo. ... Nor is the lack of a proper name unusual. Gods who are being invoked in a specific case are often left unnamed in ancient religious practice. This was common with the Egyptians (the "good god" was the King) and can also be seen in Homer, who in the Iliad invokes a Muse as "the goddess" without naming her, and also in Parmenides, who details a long instruction from a goddess who is never named. ... No, "the god" is a common locution, ... and the more that this god happened to mean personally to Socrates, the less likely that he would actually pronounce his name. ... Commentators who are unaware of this ancient religious reluctance to name the god to whom one is appealing may use the absence of the name "Apollo" in the Apology as evidence that Socrates did not believe in the god and is referring to Delphi ironically. But quite the opposite is the case. That Apollo is not named is the best evidence of the power that the identity of the god held for Socrates.

You can use Perseus to compare the original text with an English translation. He uses the form "τῷ θεῷ", which is a dative singular, so the translator's choice is correct. (I apologize for the short top-level response, but I think this is useful.)

Socrates was brought up on two charges: (1) "Refusing to acknowledge the gods recognized by the State and of introducing new and different gods," and; (2) "Corrupting the youth."

We don't have a text of the actual laws that were said to have been violated or a transcript of the prosecution's case. We have only the highly biased accounts of two of Socrates' fanbois, Plato and Xenophon.

The case against Socrates was probably founded in his relentless, unrepentant Laconophile (i.e., pro-Spartan) teachings and lifestyle. This didn't go over well in Athens, which had just emerged from an extraordinarily brutal occupation by Sparta. Some of Socrates' students were Athenian quislings who assisted the Spartans in their reign of terror. (See, e.g. Alcibiades and Plato's cousin Critias).

In broad strokes, that was the basis for the "corrupting the youth" charge. The impiety charge is rather more a "catch-all."

Socrates' studied and respected Anaxagoras, a celebrity philosopher in Athens when Socrates was a pup. Anaxagoras came up with some remarkable and novel theories for natural phenomena (e.g. the sun is an enormous mass of flaming metal, all things are infinitely divisible, celestial bodies are rocks), and was also subsequently charged and convicted of impiety.

It would seem that Socrates' defense was to disassociate himself with Anaxagoras and establish only that he is not an atheist (which is not what he was charged with).

Plato wishes to leave his reader with the false impression that Socrates was prosecuted merely for extolling people to be virtuous. "I go to die, and you to live, but which of us goes to the better lot is known to none but God," is very likely a lovely rhetorical flourish by Plato, rather than an actual quotation from Socrates.

If you're interested in the trial itself, see Prof. Doug Linder's "Famous Trials" site, http://law2.umkc.edu/faculty/projects/ftrials/socrates/socrates.HTM

Relatedly, I've been reading (in English translation) Epictetus and Seneca. The God they refer to seems pretty abstract, though Epictetus specifically denounces the (apparently) Epicurean notion of an unconcerned, completely remote God.

Did the god of the Stoics evolve, so to speak, from a more specific Greek or Roman deity to the "god of reason" that eventually gets adopted by the Enlightentment thinkers, hundreds of years later? Or is this also a projection, as the Christian theologists' gloss seems to be?

My translation is slightly different: "Now the hour to part has come. I go to die, you go to live. Which of us goes to the better lot is known to no one, except the god."

Based off traditional Socratic thinking it stands to reason he might of had an agnostic approach to God. Look at this last words: he's aware of what he doesn't know, and for Socrates this is true wisdom. Just my take, good question OP!

Socrates, Plato and most intellectual Greeks at the time did not buy into the Greek Pantheon. He may have been superficially referring to Apollo, but he almost certainly did not believe Apollo existed in any capacity remotely similar to the literary Apollo. His notion of God was more likely more akin to the more monotheistic versions of a God. One of the accusations brought against him was disregarding the "gods" meaning he was likely at the forefront of Greek life disregarding the Pantheon. But this accusation is probably not something that would be sufficient for charges to be brought against someone, rather they were tacked on to give weight to his potential guilt and paint him as a dangerous element in society (if not simply an obnoxious element)

Belief in the pantheon only became less and less prominent. By the Roman era, they were very nearly entirely reduced to literary devices. People like Ovid and so on loved using the mythology to explore literary style and stories of morality. BUt they almost certainly did not believe the Roman (or Greek) Gods existed in any capacity akin to what they are depicted.

I'm transcribing the actual passage:

Αλλά γαρ ήδη ώρα απιέναι, εμοί μεν αποθανουμένω, υμίν δε βιωσομένοις. Οπότεροι δε ημών έρχονται επι άμεινον πράγμα, άδηλον παντί πλην η τω θεώ.