[Flups]

I think I would just put BB AI and if one of the other two villains comes along I would stick it in on any non A/K flop. UTG and CO haven't shown any strength, so you're basically playing against the BB. With the dead money already in the pot I'm always sticking it in vs. a 32 BB stack with QQ (I guess if his range is something like AK, JJ+ this would be mathematically correct(?)). If you just call, you give UTG and CO good odds to call for set value and you could get in trouble when flopping an overpair, which scenario Sounded Simple did not include in his post. So, I think calling indeed is out of the question here.

If BB had a full stack calling makes more sense, although you're then basically setmining with your QQ; playing very cautiously when the flop presents all undercards and giving up when an A or K flops.