What would Q do if he had George Washington's underwear?
Assuming you mean Q from the series Star Trek: The Next Generation, I believe Q would be far more likely to bring George Washington himself to the ship. However, if only in possession of George Washington's underwear, then he would certainly use it to make a point about humanity to the crew, as well as use it to irritate Captain Picard (whom he has shown to have a long-standing interest in). Q would no doubt point out the irrational attachment that "puny humans" have to material possessions, especially those they attach historical significance to. He would likely make a game of destroying the underwear (perhaps by forcing the captain into a game by which he could save them), and in the end demonstrate that the destruction of George Washington's underwear has no affect upon history (or anything of significance). Q likes to comment on humanity, and thus the underwear would serve his purpose...with the addition of annoying the captain (which he also seems to enjoy).
If the Q you are referring to is the notorious being from Star Trek, then there are a multitude of possibilities. No doubt he would ultimately use them to gain some kind of reaction from Captain Picard, though with his devilish mind there is no telling what kind of crafty things he could accomplish.