Why do Greek gods fight?

No. Greek gods were immortal, so battle could wound, humble, or imprison them, but not kill them. Ares was wounded by the mortal Diomedes and fled to Olympus to heal, while the defeated Titans were locked in Tartarus. That is why divine wars ended in punishment or exile rather than death.

The Titanomachy was a ten-year war between the Olympian gods, led by Zeus, and the older Titans, led by Cronus. Zeus rebelled after freeing his swallowed siblings. The Olympians won, locked the Titans in Tartarus, and divided the world, with Zeus taking the sky, Poseidon the sea, and Hades the underworld.

The Apple of Discord was a golden apple marked for the fairest, left at a wedding by the goddess Eris, who was not invited. Hera, Athena, and Aphrodite all claimed it, and the Trojan prince Paris had to choose. His choice of Aphrodite, who promised him Helen, helped set off the Trojan War.

Zeus, as king of the gods, usually settled them, though his rulings often failed. He decided how Adonis's year was split between Aphrodite and Persephone, had Hermes bring the goddesses to Paris for judgment, and forbade the gods from meddling in the Trojan War. They ignored that ban, so the quarrels continued.