Are we betting flop/turn

50plo - clubgg - min 35 vpip

Are we betting flop/turn?
Or we just checking back giving up unless Q hits?

Hero (CO): 149.86 BB
BTN: 103.98 BB (VPIP: 37.14, PFR: 21.43, 3Bet Preflop: 0.00, Hands: 89)
SB: 75.44 BB (VPIP: 50.00, PFR: 7.14, 3Bet Preflop: 5.88, Hands: 38)
BB: 58.78 BB (VPIP: 66.67, PFR: 33.33, 3Bet Preflop: 0.00, Hands: 4)
UTG: 91 BB (VPIP: 26.09, PFR: 8.70, 3Bet Preflop: 0.00, Hands: 31)
UTG+1: 22.5 BB (VPIP: 81.82, PFR: 6.82, 3Bet Preflop: 0.00, Hands: 59)
MP: 169.3 BB (VPIP: 66.67, PFR: 14.81, 3Bet Preflop: 11.76, Hands: 34)

SB posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has A♠ K♥ Q♦ Q♣

fold, UTG+1 calls 1 BB, MP calls 1 BB, Hero raises to 5.5 BB, fold, fold, BB calls 4.5 BB, UTG+1 calls 4.5 BB, MP calls 4.5 BB

Flop: (22.5 BB, 4 players) 4♦ 9♠ 2♠
BB checks, UTG+1 checks, MP checks, Hero checks

Turn: (22.5 BB, 4 players) 9♣
BB checks, UTG+1 checks, MP checks
Hero???