In Book 3 Telemachos lands on Pylos, the home of the great Trojan War hero Nestor. Telemachus is welcomed by Nestor, and then Telemachus asks of news of his own father, Odysseus. Nestor doesn't have much specific knowledge of Odysseus after the Trojan War, but Nestor does give Telemachus encouragement in for the matter of his mother's menacing suitors:
Friend, since you have recalled to my mind these matters and spoken,
men do talk of the suitors who seek your mother, the many
there are in your house who against your will are devising you evils.
Tell me, are you subdued quite willingly, or do the people
hate you there in your land in response to a god's injunction?
Who is to know whether he will return some day and requite their
violence, either alone or with all the Achaians to aid him?
For if bright-eyed Athena so wanted to favor and love you
as at the tim she cared for Odysseus the glorious hero,
when in the land of the Trojans Achaians were suffering sorrows --
never have I seen gods so openly showing their favor
as beside him stood openly, aiding him, Pallas Athena--
if in her heart she wanted to love you and care for you that much,
then at least some of those men would surely forget aout marriage? (III. 210-24)
Nestor tells Telemachus and Peisistratos to go to Sparta, where King Menelaus and Queen Helen may have more news of Odysseus. Helen recognizes Telemachus from his resemblance to Odysseus. Peisistratos reveals Telemachus' identity, and they tell Helen and Menelaus the reason for their visit. They all weep, mourning the loss of all the Achaians at Troy. They become so sad that Helen resorts to giving them a drug to curb their sorrow.
Then other things were devised y Helen the daughter of great Zeus:
straightaway into the wine they were drinking she cast an elixir
banishing sorrow and anger and ridding the mind of all evils.
He who swallowed the potion when it had been mixed in the wine bowl
would not shed any tears from his cheeks fo rthe day that he drank it, (IV. 219-223)
After this Menelaos tells Telemachus all he knows about what happened after the Trojan War to Odysseus. The last Menelaos had heard of Odysseus, he was alive.
Source: Homer. The Odyssey. Rodney Merrill, trans. Ann Arbor, Michigan: University of Michigan Press, 2006.