No. Paintings are furniture items. Redd sells them, whenever his tent is in town, but you'll want to search for one of the threads/wikis that shows which items are real and which are forgeries, because you can't donate forgeries to the museum.
Villagers will also sometimes step up and give/sell/trade you paintings, but it's not all that common. I've gotten one (my one real painting) so far that way. Colton's my bro, he's got my back. --- Please insert witty quote here.