In all of Homer there is only one reference to writing. The reason is that the Iliad and Odyssey were oral poems and pre-dated alphabetical writing and perhaps were mainly composed in their present form after the fall of the Mycenaean civilisation and so were after Linear-B.
Actually I don’t buy this and think the reason has more to do with fish. One thing even Plato noted about Homer was that nobody ate fish, this was despite the fact that the Greeks in the Iliad were camped on the sea and Odysseus in his wanderings would have had plenty of opportunities to dangle a fishing rod over the side of his boat. The Greeks must have eaten fish and even Homer must have occasionally tucked into a plate of red mullet perhaps accompanied by a glass of Phalerian.
The fact is that eating fish just didn’t fit the heroic ideal and I think the same could be said for writing. Let us suppose that writing was at first mostly used for accounting such as we know Linear B was used. Imagine Homer describing Agamemnon striding onto the battlefield as Agamemnon, great at accounting or saying that the Greeks were grievously harmed by Achilles refusing to do any more double entry bookkeeping after Agamemnon had seized his favourite pencil.
So what do we make of the one allusion to writing in Homer. The background to this is that the wife of Proitos, king of Tiryns, wanted to make love to Bellerophon. When he refused, she accused him of trying to force her. So Proitos sends him to his father-in-law in Lycia hoping he will get rid of him.
κτεῖναι μέν ῥ᾽ ἀλέεινε, σεβάσσατο γὰρ τό γε θυμῷ,
πέμπε δέ μιν Λυκίην δέ, πόρεν δ᾽ ὅ γε σήματα λυγρὰ
γράψας ἐν πίνακι πτυκτῷ θυμοφθόρα πολλά,
δεῖξαι δ᾽ ἠνώγειν ᾧ πενθερῷ ὄφρ᾽ ἀπόλοιτo
Proitos was reluctant to kill him for he was ashamed to do that, but he sends him to Lycia and gave him grievous symbols to take, having inscribed many things to do with death in a folded tablet and told him to show it to his father-in-law so that he would lose his life.
Many volumes have been written about what the symbols were. Some of the explanations are.
It was written in Linear B. Was Linear B used in this way? Seeing as most of the remains of this script have fortuitously survived from palaces that have been set on fire thus hardening the clay, and are to to with administration, we can have no idea what other purposes it may have had and what other media it could have been used on that haven’t survived. So it could have been used for this purpose.
It wasn’t writing. The symbols were pictures. It would be a difficult concept to put across in pictures. This man tried to rape my wife, your daughter, so please kill him. This is an unlikely explanation.
This story was imported from the East and the script was some form of cuneiform. Possibly.
It was a late addition to the poem added once alphabetical writing had become established in Greece. In order not to introduce something so unheroic as writing into the poem, he refers vaguely to symbols or signs. Again I think this is possible.
Personally I don’t think we will ever know and most guesses are as good as each other.
Bellerophon managed to survive several attempts to kill him by the king of Lycia including being sent to slay the Chimaera and eventually married the king’s daughter and was given half the kingdom but in later life went mad.