What am I supposed to do here?

Should I have bet more? My turn bet was small because I was trying to induce a raise. At the time I'm not sure what I was expecting from villain. Was this board just too scary for him to do anything other than flat call?

How would you have played this?

Party Poker No-Limit Hold'em, $ BB (8 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

saw flop|saw showdown

UTG ($24.65)
UTG+1 ($26.84)
Hero ($24.75)
MP2 ($24.80)
CO ($43.66)
Button ($17.68)
SB ($23.90)
BB ($25)

Preflop: Hero is MP1 with J, Q. SB posts a blind of $0.10.
UTG calls $0.25, UTG+1 calls $0.25, Hero raises to $1, MP2 calls $1, 4 folds, UTG folds, UTG+1 calls $0.75.

Flop: ($3.60) J, Q, Q (3 players)
UTG+1 checks, Hero bets $1, MP2 calls $1, UTG+1 folds.

Turn: ($5.60) T (2 players)
Hero bets $2, MP2 calls $2.

River: ($9.60) K (2 players)
Hero bets $6, MP2 calls $6.

Final Pot: $21.60

Results in white below:
Hero has Jh Qh (full house, queens full of jacks).
MP2 has Ad As (straight, ace high).
Outcome: Hero wins $21.60.

Other than betting a bit more on each street - half a pot at least, most people with good draws will call that - you could have pushed on the river. There's a strong chance any ace would call any bet here despite the flush on the board.