QQ underpair in 3bet pot OOP - 100NL

[Dory]

Sorry if this is kind of basic but I felt a bit lost.

When a good player calls your 3 bet from the button and an overcard flops, what is the line?

Check-call and check-fold seem wrong on the flop, so I felt I had to bet, but then I feel like he can push me off a hand.

Is this just standard, and I just have to check the turn to him with AK, AA and sets?

Party Poker - No Limit Hold'em Cash Game - $0.50/$1 Blinds - 5 Players - (LegoPoker Hand History Converter)

SB: $201.34
Hero (BB): $142.25
UTG: $237.99
CO: $124.41
BTN: $121.48

Reads: <font color="blue">Villain is a very solid 20/15 player. I have a fair amount of respect for his game, and generally try not to tangle with him.</font>

Preflop: Hero is dealt Q[img]/images/graemlins/diamond.gif[/img] Q[img]/images/graemlins/spade.gif[/img] (5 Players)
2 folds, <font color="red">BTN raises to $4.00</font>, SB folds, <font color="red">Hero raises to $14.00</font>, BTN calls $10.00

Flop: ($28.50) 4[img]/images/graemlins/heart.gif[/img] K[img]/images/graemlins/spade.gif[/img] 7[img]/images/graemlins/spade.gif[/img] (2 Players)
<font color="red">Hero bets $18.00</font>, BTN calls $18.00

Turn: ($64.50) 6[img]/images/graemlins/heart.gif[/img] (2 Players)
Hero checks, <font color="red">BTN bets $40.00</font>, Hero folds

Pot Size: $104.50