Start mucking QQ preflop imo.

Turn is tough call, I hate folding this to a 1/4 pot bet. If you think this is a bluff at least 20% of the time and he's incapable of firing a second barrel on the river without an ace, call. River is weird, you've shown interest in staying in the hand, yet he's still betting tiny (missed flush, cheap bluff??) I'd want to spite call for cheap just to see the ace, but I think the best play is just fold.