I would assume Buttons hand to be somehow related to the Q.
If his hand is good on the flop, the only reason not to charge you all for another bet would be to set up a check-raise on the turn. But he bets out on the turn, which seems to indicate thats not the case.

So either he is not thinking, making some sort of play that I really dont understand or being helped by the Q.

His play pf play doesnt indicate QQ, and his plays on the later streets is completely insane with only one pair of Q?
I would guess caught two pairs when the Q came, and maybe boated up on river.

The Button I would expect to be playing another set, unless he is grossly overplaying his cards.