When Athena departs from Telemachus and Nestor how does she reveal that she is a Goddess?
In the early sections of the Odyssey, Athena appears to Telemachus in disguise as Mentor. As Mentor, she gives Telemachus the confidence to speak to Nestor about his father, Odysseus. Nestor relates what he knows (which is little), but bids Telemachus seek his father with the same zeal Orestes took in avenging Agamemnon’s death. Nestor sends Telemachus to Sparta to seek information about his father, and the wise old king sends his son, Pisistratus, as a companion. Athena departs following these events.
When Athena departs from Telemachus and Nestor, she reveals herself to be a goddess by putting off her disguise and flying away in the likeness of a vulture (or eagle, depending on your translation). As the Richard Lattimore translation relates in Book 3, lines 371-377 (the spellings in his translation differ from some spellings in other translations):
So speaking, gray-eyed Athene went away in the likeness
of a vulture, amazement seized on all the Achaians,
and the old man was amazed at what his eyes saw. He took
Telemachos by the hand and spoke a word to him and named him:
‘Dear friend, I have no thought that you will turn out mean and cowardly
if, when you are so young, the gods go with you and guide you