Whether Plato's character of Socrates or Plato himself believed in a God or many Gods is not perfectly clear. Additionally, we can't ascribe any sort of belief to the historical Socrates; we just don't know enough about his life that doesn't come from Plato or Xenophon. For the rest of this answer I'll say "Socrates" instead of "Plato's character of Socrates" for the sake of brevity.
The Greek word Plato used in the original text is "θεός" (theos) which is translated to "God" or "the deity". However "θεοί" which is the plural form of "θεός" is also used. The translators did not change Plato's plural use to the singular use to reflect their own monotheistic views, Plato does indeed use both in the text. Here is a chart for the declension of theos via Wikitonary:
For example, in the Apologia, we have these two sections of text.
οὗτοι, ὦ ἄνδρες Ἀθηναῖοι, οἱ ταύτην τὴν φήμην κατασκεδάσαντες, οἱ δεινοί εἰσίν μου κατήγοροι: οἱ γὰρ ἀκούοντες ἡγοῦνται τοὺς ταῦτα ζητοῦντας οὐδὲ θεοὺς νομίζειν. - 18ξ (θεοὺς - plural accusative)
These, men of Athens, who have spread abroad this report, are my dangerous enemies. For those who hear them think that men who investigate these matters do not even believe in Gods. 18c
ὅμως τοῦτο μὲν ἴτω ὅπῃ τῷ θεῷ φίλον, τῷ δὲ νόμῳ πειστέον καὶ ἀπολογητέον. 19α (θεῷ - singular dative)
But nevertheless, let this be as is pleasing to God, the law must be obeyed and I must make a defence. 19a
Before we discuss whether or not Socrates believes in a monotheistic or polytheistic god, we should state that it is clear is that Socrates and Plato rejected the Homeric image of the Greek pantheon. Of course, one of the main charges against Socrates was that he did not believe in the Gods of Athens and he preached heresy to the youth of the city. Book II of the Republic is devoted to Socrate's discussion that the real Gods must be just and honest and myths that portray them as being petty and dishonest are false and should not be taught to children. From the SEP:
Evidence for irreverence was of two types: Socrates did not believe in the gods of the Athenians (indeed, he had said on many occasions that the gods do not lie or do other wicked things, whereas the Olympian gods of the poets and the city were quarrelsome and vindictive); Socrates introduced new divinities (indeed, he insisted that his daimonion had spoken to him since childhood).
In this context daimonion is the Greek word for demon which in ancient Greek culture had a much different meaning than the meaning in, say, Christian theological contexts. From the Online Etymology Dictionary's entry for demon:
c. 1200, from Latin daemon "spirit," from Greek daimon "deity, divine power; lesser god; guiding spirit, tutelary deity."
The original mythological sense is sometimes written daemon for purposes of distinction. The Demon of Socrates was a daimonion, a "divine principle or inward oracle." His accusers, and later the Church Fathers, however, represented this otherwise.
So now on to the question of whether Socrates believed in one God or many Gods. The generally accepted answer is that Socrates believed in an ambiguous monotheistic God. This view is similar to (and I hope I do not offend anyone by giving a simplistic view of) the Brahman in Hinduism. Socrates speaks amply of paying tributes to the Gods and that the Gods are virtuous, and yet he also speaks of God being the eternal source of everything that is good. Similarly to how Hindu Gods and Goddesses are aspects of the same singular deity, the Brahman, Socrates believes that the Gods and Goddesses are aspects of the same monotheistic deity. In the original text Plato uses both singular and plural forms of "Theos" much like ancient Hindu texts use the singular and plural form of "Deva" (Sanskrit word for God). Socrates believed that his inner daimonion was his, and all of our, means of communication with this deity. On a monotheistic view it is very similar to how Christian theology considers the Holy Spirit in relation to the Godhead.
However, there is an alternative. It should be pointed out that Socrates specifically refers to Apollo (Apollo is the specific God whom the oracle of Delphi communicated with) on multiple occasions.
For the word which I will speak is not mine. 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.
In the Apologia Socrates tells the court that the Oracle at Delphi told him, through a friend, that Socrates was to be a great philosopher. The argument is made that when Socrates refers to a singular God he is referring to Apollo, as Apollo was the specific God who guided him to his wisdom. The general response to this from the scholars who believe that Socrates is monotheistic is that it is clearly different when Socrates refers to one specific God who is Apollo and one specific God who is the one deity that exists.
For the sake of more examples to illustrate why people believe that Socrates was monotheistic even though he mentions both the Gods and God, see this excerpt from the Republic:
κομιδῇ ἄρα ὁ θεὸς ἁπλοῦν καὶ ἀληθὲς ἔν τε ἔργῳ καὶ λόγῳ, καὶ οὔτε αὐτὸς μεθίσταται οὔτε ἄλλους ἐξαπατᾷ, οὔτε κατὰ φαντασίας οὔτε κατὰ λόγους οὔτε κατὰ σημείων πομπάς, οὔθ᾽ ὕπαρ οὐδ᾽ ὄναρ. 382ε (θεὸς - singular nominative)
“Then God is altogether simple and true in deed and word, and neither changes himself nor deceives others by visions or words or the sending of signs in waking or in dreams.” 382e
συγχωρεῖς ἄρα, ἔφην, τοῦτον δεύτερον τύπον εἶναι ἐν ᾧ δεῖ περὶ θεῶν καὶ λέγειν καὶ ποιεῖν, ὡς μήτε αὐτοὺς γόητας ὄντας τῷ μεταβάλλειν ἑαυτοὺς μήτε ἡμᾶς ψεύδεσι παράγειν ἐν λόγῳ ἢ ἐν ἔργῳ; 383α (θεῶν - plural genitive)
“You concur then,” I said, “this as our second norm or canon for speech and poetry about the gods,—that they are neither wizards in shape-shifting nor do they mislead us by falsehoods in words or deed?” 383a
Finally, for further textual references, the dialogs in which Socrates discusses what the Gods are like in respect to this conversation are Euthyphro, Republic book II, Phaedrus, and of course the Apologia.