The strongest candidate for a god of writing would be Hermes, the god of - amongst other things - language:
Also the Guide, the Slayer of Argus, contrived within her lies and crafty words and a deceitful nature at the will of loud thundering Zeus, and the Herald of the gods put speech in her. And he called this woman Pandora, because all they who dwelt on Olympus gave each a gift, a plague to men who eat bread.
Source: Hesiod, Works and Days, lines 75-80
Plato elaborates on Hermes' association with language, in Cratylus:
I will do so, but first one more god. I want to ask you about Hermes, since Cratylus says I am not Hermogenes (son of Hermes). Let us investigate the name of Hermes, to find out whether there is anything in what he says.
Well then, this name “Hermes” seems to me to have to do with speech; he is an interpreter (ἡρμηνεύς) and a messenger, is wily and deceptive in speech, and is oratorical. All this activity is concerned with the power of speech. Now, as I said before, εἴρειν denotes the use of speech; moreover, Homer often uses the word ἐμήσατο, which means “contrive.” From these two words, then, the lawgiver imposes upon us the name of this god who contrived speech and the use of speech—εἴρειν means “speak” and tells us: “Ye human beings, he who contrived speech (εἴρειν ἐμήσατο) ought to be called Eiremes by you.” We, however, have beautified the name, as we imagine, and call him Hermes. Iris also seems to have got her name from εἴρειν, because she is a messenger.
Source Plato, Cratylus, p.408
Hermes was the messenger of the gods, and naturally writing would be essential to him. He isn't however credited with inventing writing, at least not until Roman times:
 CXLIII. PHORONEUS
Inachus, son of Oceanus, begat Phoroneus by his sister Argia, and he is said to have been the first of mortals to rule. Men for many centuries before lived without town or laws, speaking one tongue under the rule of Jove. But after Mercury had explained the languages of men (when he is called ermeneutes, “interpreter,” for Mercury in Greek is called Hermes; he too, divided the nations), then discord arose among mortals, which was not pleasing to Jove. And so he gave over the first rule to Phoroneus, because he was first to make offerings to Juno.
 CCLXXVII. FIRST INVENTORS
The Parcae, Clotho, Lachesis, and Atropos invented seven Greek letters - A B H T I Y.
Others say that Mercury invented them from the flight of cranes, which, when they fly, form letters.
Palamedes, too, son of Nauplius, invented eleven letters; Simonides, too, invented four letters – Ó E Z PH; Epicharmus of Sicily, two - P and PS.
The Greek letters Mercury is said to have brought to Egypt, and from Egypt Cadmus took them to Greece. Cadmus in exile from Arcadia, took them to Italy, and his mother Carmenta changed them to Latin to the number of 15.
Apollo on the lyre added the rest.
Source: The Myths of Hyginus, translated and edited by Mary Grant
Hyginus belief that Hermes brought the Greek letters from Egypt is interesting because Hermes would some times be identified with Thoth, who the Egyptians did credit with inventing writing.
That said, the Greeks probably didn't think of writing as a product of divine inspiration; in fact, they were aware their alphabet derived from the Phoenician one:
These Phoenicians who came with Cadmus and of whom the Gephyraeans were a part brought with them to Hellas, among many other kinds of learning, the alphabet, which had been unknown before this, I think, to the Greeks. As time went on the sound and the form of the letters were changed.
Source: Herodotus, The Histories, Book 5, chapter 58, section 1