9-man Full Tilt 1$ turbo SnG JJ postflop

Flop:I think he rarely hits a Q so I bet about 2/3 of the pot to protect from draws.
Turn:I barrel the turn because of the OESD
River:I check behind in case he slowplays a monster and wants to induce a bluff

I would also bet the flop and turn, maybe slightly bigger on the turn. On the river you could consider a bet to get value out of for example a nine or ten, or some worse bluff catcher, folding to a raise (when we are more or less always beat). Those middling cards must figure quite a bit in a BB calling range, whereas bigger pocket pairs and better queens might be expected to have 3 bet preflop.

A thin value bet would be indeed a good option, but that depends a lot with what kind of player we are dealing with, but I expect most <$1 players are capable to call to wide here. So a value bet on river would be fine here.

Cheers,
SDK1987