Trying not to scare them off

I don't have any reads because I had only just arrived at the table but most of the players seemed to be playing fairly standard except one who was quite LAG. The point that when I got to the river I wasn't sure how big a bet to place that wouldn't scare Villain off but would get me the most. How should I have played this?

Cassava Poker $0.01/$0.02 No Limit Hold'em - 8 players
The Official 2+2 Hand Converter Powered By DeucesCracked.com

CO: $2.01
BTN: $0.37
SB: $5.00
BB: $2.38
UTG: $3.51
UTG+1: $0.80
MP1: $2.28
Heroine (MP2): $1.51

Pre Flop: ($0.03) Heroine is MP2 with 4 Q
3 folds, Heroine calls $0.02, 3 folds, BB raises to $0.04, Heroine calls $0.02

Flop: ($0.09) 6 8 9 (2 players)
BB bets $0.09, Heroine calls $0.09

Turn: ($0.27) 4 (2 players)
BB checks, Heroine checks

River: ($0.27) 5 (2 players)
BB checks, Heroine ?