Your primary question is perhaps a better question for the "Short Answers to Simple Questions" thread, because regrettably the answer is that they did not. Classicists have discovered only one reference to the Hittites in Greek literature, a passage in Book 11 of the Odyssey in which Odysseus is speaking to Achilles in the netherworld.

I couldn't recount or name all the victims [Neoptolemus, the son of Achilles] slew,

all the many men he dispatched when fighting for the Argives -

but one great man, Telephos' son, he felled with the bronze:

the hero Eurypylos, and with him a crowd of his Keteian

comrades all perished because of a woman's gifts.

The Keteians have been identified with the Hittites, and Telephos is almost certainly a Greek version of the Hittite royal and divine name Telepinu.

More can be said about the impact of Hittite mythology, however. I'll write about two of the most obvious parallels, the Illuyanka myths and the Kumarbi cycle.

Illuyanka and the Storm God

Two versions of the Illuyanka myth have survived.

(1) Version 1

The giant serpent Illuyanka and the Storm God fought a fierce battle in the town of Kiškilušša in Anatolia. Illuyanka eventually defeated the Storm God, who crept away in defeat. Devising a plan to defeat the serpent, the Storm God summoned the other Hittite gods to a feast. His daughter Inara, the goddess of wild animals, prepared a magnificent feast with a large quantity of food, wine, and beer. Inara then visited the town of Ziggaratta and invited a man named Ḫupašiya to the feast. Ḫupašiya agreed to come to the feast, but only on one condition.

Ḫupašiya replied as follows to Inara: "If I may sleep with you, then I will come and perform your heart's desire." So he slept with her.

After dressing herself in fine clothes and concealing Ḫupašiya, Inara visited the lair of Illuyanka and invited him to her feast. Delighted to be invited, Illuyanka appeared at the feast with his children and proceeded to take advantage of the open bar. Once the serpent was thoroughly intoxicated, the mortal Ḫupašiya jumps out of hiding and tied up Illuyanka. The Storm God, delighted to have his enemy at his mercy, slayed the helpless serpent.

(2) Version 2

In this version of the myth, Illuyanka not only defeated the Storm God but stole the god’s eyes and heart. The Storm God was blinded but still alive, and he quickly hatched a plan to regain his lost body parts. The god married the daughter of a poor man, and together they produced a son. Once the son of the Storm God reached manhood, his father encouraged him to marry the daughter of Illuyanka. In Hittite society the parents of the bride paid a dowry to the groom's family, and so the Storm God told his son to ask for his heart and eyes. His son dutifully asked for his father’s heart and eyes from Illuyanka, and they were returned to him. The restored Storm God attacked Illuyanka once more, and this time the snake was killed. The son of the Storm God, full of grief and remorse at inadvertently causing the death of his wife’s father, asked his father to kill him as well.

The Illuyanka myths shares several similarities with the battle between Zeus and Typhon in Greek mythology, particularly the neutralization of a god by removing his body parts. In the battle between Zeus and Typhon, Typhon seized Zeus in his coils and removed the sinews from the hands and feet of the god. Thanks to the intervention of Hermes and Aegipan, Zeus was able to restore himself and defeat Typhon.

Alalu, Anu, Kumarbi, and the Storm God

As the tale begins, the god Alalu was the king of the gods. He ruled wisely for nine years, and for nine years the other gods bowed to him and obeyed his wishes. Anu, his most important servant, was in charge of tending to the king and providing him with cups of wine. Anu grew dissatisfied with his position as servant after nine years and attacked Alalu. After a fierce battle between the king and his former servant, the servant emerged victorious as the new king of the gods, and the defeated Alalu fled to the underworld.

Anu ruled for nine years, and for nine years his servant Kumarbi brought him cups of wine. Kumarbi was no less ambitious than Anu, however, and so he attacked and dethroned Anu. As Anu attempted to flee to the sky, Kumarbi seized the god and bit off and swallowed his genitals. Before escaping to the sky, Anu informed Kumarbi that he would have three children: the Storm God, the Storm God’s brother Tašmišu, and Aranzaḫ, the Tigris River. Kumarbi understandably panicked when he heard this news and attempted to spit out Anu's genitals in a failed attempt to avoid the prophecy. Suffering from pain from the birthing process, Kumarbi summoned Ea, the god of wisdom, to ease the pain. Kumarbi was apparently aware of the danger the baby Storm God posed to his reign, and so he asked Ea to give him the baby Storm God so that he could consume it. The text becomes quite fragmentary at this point, but it seems the gods hid the baby and gave Kumarbi a rock to eat instead. In any case, the Storm God eventually overthrew his father, became king, and exiled Kumarbi.

The cycle is obviously strikingly similar to the cycle of succession in Hesiod's Theogony. According to Hesiod, Ouranos was the father of the first generation of Titans. Several of his monstrous children were imprisoned within Gaia, the earth, who grew to despise him for the pain he caused her. Gaia produced a sickle and urged her children to free her of Ouranos. Her son Kronos stepped forward and agreed to do the deed. Taking the sickle, Kronos ambushed and castrated his father. The blood caused the birth of the Erinyes (Furies), Giants, and Meliae (ash tree nymphs), and the genitals, thrown into the sea, washed ashore on Cyprus as a white foam, from which arose the goddess Aphrodite.

Although Kronos produced children with Rhea, he was no less paranoid than Kumarbi and consumed each child to prevent one of his offspring from overthrowing him. Rhea grew tired of his tyranny, however, and secretly gave birth to Zeus on Crete, after which she presented Kronos with a swaddled stone in place of his son. Once the baby Zeus had grown into adulthood, he freed his siblings and defeated Kronos.