You’ll trigger this quest on the 13th of Winter in your second year. Check your mail when you wake up on that day and you’ll find a letter from Willy waiting. He wants you to catch a Lingcod, a type of fish that only appears during the Winter.
Lingcods can appear in a few places, but the most convenient fishing point for you is the river that runs through the south of central Pelican Town. Park yourself somewhere south of the houses and start fishing. You should have the best fishing rod you can afford - Iridium is the obvious choice, though Fiberglass can do as well - and some Bait to tempt the thing to your line. Lingcods are pretty testy and can give you a bit of a fight, so having a Fishing stat of five or above will make your job significantly easier.
Take the Lingcod to Willy, who lives at his shack on the docks of the beach. He’ll give you 550 gold as recompense via your Journal.