It's villain's second hand out of 15 that he's played since he sat down. I have no reads. In retrospect I should have 3bet this pot but I erred on the side of caution since I had no history with villain. Is this one of those spots where we can bet/fold the turn in order to narrow villain's range? If I check call turn for say 4$ or so, I feel river would be a difficult decision faced with a shove for villain's last 13$ or so since at that point I'd basically want to check it down. Thoughts?

No-Limit Hold'em, $0.20 BB (6 handed) - Hold'em Manager Converter Tool from FlopTurnRiver.com

saw flop

UTG ($9.54)
MP ($29.94)
CO ($28.95)
Button ($19.90)
Hero (SB) ($19.20)
BB ($19.83)

Preflop: Hero is SB with Q, A
2 folds, CO calls $0.20, Button bets $1, Hero calls $0.90, 2 folds

Flop: ($2.40) Q, 2, J (2 players)
Hero checks, Button bets $2, Hero calls $2

Turn: ($6.40) 6 (2 players)
Hero ???