What do the filled baskets do? Are they just donation fodder, or do they have any other event-related uses?

they're Holiday Items, you so can give them to Quentin for goodies, and you can feed them to your own pets if you're going for Gourmand.

I just found a basket how do you fill them?

take them to the Farmers Market in Veta. They should show on the bottom of that page for you to click.

How do I give to Quentin? Thank you :)

You can't unless it comes up in a quest for them. He just meant that the baskets are the right rarity so you could hold onto them for that if you wanted

Ah ok...thank you so much :)