Cheating on one's significant other is common in Greek Myth. I have only found four gods that remain faithful:
Hades: The Most Loyal Greek God
The Greek god Hades is comparatively a better husband than his peer gods. Whilst Zeus and Poseidon – Hades' brothers – are widely known for their affairs, Hades remained loyal to Persephone. However, there are two cases in which Hades' loyalty is questioned: Minthe and Leuce.
Despite the affairs of Zeus, Hera remained faithful and was one of the few Gods or Goddesses in Greek mythology who was monogamous.
Hera, as the goddess of marriage, was the only Greek god/goddess to remain faithful to her spouse, Zeus. It was because of this fact that she was would get so angry when he cheated on her. Hera wanted the same treatment that she gave Zeus, the same fidelity and faithfulness.
Unfortunately, Zeus constantly cheats on Hera and he has done it over a hundred times, but in the end Hera always forgives him.
Ganymede (or Ganymedes) was a young man from Troy. His beauty was unparalleled, and for that reason, Zeus abducted and brought him to Olympus to serve as his cupbearer and lover. Ganymede's myth is an important step in queer history, but there is also a dark side to the story.
One version of Persephone's story told by the Roman poet Ovid might suggest she had grown some feelings of affection for Hades in spite of everything. In Ovid's famous text Metamorphosis, Hades has an affair with a young Nymph named Minthe.
Zeus was worshiped far and wide across the Greek world, including at festivals such as the Olympic Games. His legacy as the greatest of gods also meant that he became the favored deity of great leaders in the ancient world.
PISTIS was the personified spirit (daimona) of trust, honesty and good faith.
Hestia in Greek Mythology
Hestia was regarded as one of the kindest and most compassionate amongst all the Gods. Perhaps the first example of a benign God or Goddess.
Aphrodite later and of her own volition had an affair with Zeus, but his jealous wife Hera laid her hands upon the belly of the goddess and cursed their offspring with malformity. Their child was the ugly god Priapos.
General. Like the rest of the Gods, Hades does not have a set sexuality. However, he remains in a monogamous relationship (unlike most Gods).
Zeus finally became enamored of the goddess who was to become his permanent wife — Hera.
Astraea was the goddess of innocence in Greek mythology, daughter of the Titans Astraeus, god of dusk, and Eos, goddess of dawn.
Like Athena and Artemis, Hestia elected to never marry and remain an eternal virgin goddess instead, forever tending to the hearth of Olympus. Despite her limited mythology, Hestia remained a very important goddess in ancient Greek society.
In Greek mythology, Phthonus (/ˈθoʊnəs/; Ancient Greek: Φθόνος Phthónos), or sometimes Zelus, was the personification of jealousy and envy, most prominently in matters of romance. In Nonnus's Dionysiaca, he is by proxy the cause of Semele's death, having informed Hera of Zeus's affair with the princess.
Hypochondria, Least Powerful of the Greek Gods ably distinguishes real pain from fake pain and identifies the pain of having to translate the story about your body's pain so others may understand you.
In ancient Athens, Eleos (Ancient Greek Ἔλεος m.) or Elea was the personification of mercy, clemency, compassion and pity – the counterpart of the Roman goddess Clementia. Pausanias described her as "among all the gods the most useful to human life in all its vicissitudes."
Regardless of her birth into existence, Aphrodite was born the goddess of love and was known for being the most beautiful Greek goddess.
Proteus knew all things—past, present, and future—but disliked divulging what he knew. Those who wished to consult him had first to surprise and bind him during his noonday slumber.
DOLOS (Dolus) was the personified spirit (daimon) of trickery, cunning deception, craftiness, treachery and guile. He was an apprentice of the Titan Prometheus and a companion of the Pseudologoi (Lies).
She turned him down, as Zeus had already proposed to her and by accepting his proposal she would be Queen of the Gods. Despite this, the two still held a candle for each other. Due to Zeus' constant infidelity during their marriage, Hera began a long-term, on and off affair with Hades.
Sisyphus is a figure in Homer's Iliad and other works of Greek mythology. He is reputed to be the founder of the Isthmian Games and is a trickster who receives eternal punishment for trying to cheat Death.
In the Orphic myths, the maiden goddess Persephone was seduced by Zeus in the guise of a serpent. She bore him a son, the godling Zagreus, who, when Zeus placed him upon the throne of heaven, was attacked and dismembered by the Titanes. His heart was recovered and he was reborn through Semele as the god Dionysos.