You’ll trigger this quest on the 6th of Fall. Check your mail and you’ll find a letter from Gus, asking for Lobster. Sounds like a plan, boss.
Lobsters are pulled from the ocean via Crab Pots, which require a Fishing level of three, along with 40 Wood and three Iron Bars. (Trappers get Crab Pots for 25 Wood and two Copper Bars, if you went that route.) You’ll also need some Bait, which you can purchase from the fishing shop for a scant five gold apiece.
Make a few Crab Pots and toss them into the water near Willy’s home, then keep checking them each day. You’ll eventually luck out and get the Lobster Gus wants. Snag it and take it to the restauranteur at the Stardrop Saloon, where he spends the majority of his days. He’ll give you 500 gold for your trouble.