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The Greek gods could definitely age.

Zeus, unlike his most famous daughter Athena, who sprang from his head fully grown and armored, is an infant at his own birth:

Enraged at this, Rhea repaired to Crete, when she was big with Zeus, and brought him forth in a cave of Dicte. She gave him to the Curetes and to the nymphs Adrastia and Ida, daughters of Melisseus, to nurse ... these nymphs fed the child on the milk of Amalthea, and the Curetes in arms guarded the babe in the cave ... Rhea wrapped a stone in swaddling clothes and gave it to Cronus to swallow, as if it were the newborn child.
SOURCE: Apollodorus 1.1.6 ff.


Hermes is the best example of a Greek god beginning as a child and ultimately having adult form:

Maia, the eldest, as the fruit of her intercourse with Zeus, gave birth to Hermes in a cave of Cyllene. He was laid in swaddling-bands on the winnowing fan, but he slipped out and made his way to Pieria and stole the kine which Apollo was herding.
SOURCE: Apollodorus 3.10.2

Hermes was definitively in child form:

...Apollo came to Pylus in search of the kine, and he questioned the inhabitants. They said that they had seen a boy driving cattle ... Apollo came to Maia at Cyllene and accused Hermes. But she showed him the child in his swaddling-bands.
SOURCE: ibid.

Hermes, of course, grew into adulthood, as he is commonly depicted as a kouros (young man), such as in this statue.

Although some of the Greek gods age into their final, adult forms, there are no examples of the Greek gods getting old or dying of old age, so far as I am aware.

Thus, although the Greek gods could age, they are nevertheless eternal, ageless, and immortal.