You’ll trigger this quest on the 8th of Fall when a piece of mail arrives at your farm. Check it and you’ll discover that Linus has lost his Blackberry Basket somewhere. He wants you to get it back post-haste.
Fortunately, finding it is relatively easy - or, at the very least, it’s relatively quick. Leave your farm, heading towards Pelican Town, then swing north. You’ll find the bus stop. It’s currently (probably) out of order. Head west from the bus, though, and you can make the trek on foot to find a tunnel. The Berry Basket is sitting on the side of the road beside the tunnel.
Take the Berry Basket back to Linus. He lives in the Tent to the north of Pelican Town, between the Carpenter’s Shop and the Mines, and usually hangs out in the general area. Linus doesn’t give you anything tangible for finding the Basket, but you will earn a big affection boost with the poor man.