So my turn barrel is shocking but ignore that for now. My opponent represents very few hands when he leads out Qxclubs, AT, perhaps QJo. But I wouldn't expect too many flush draws to bluff this board considering I've bet both flop and turn and the river paired the turn.

SB: 103.9 BB (VPIP: 18.12, PFR: 13.68, 3Bet Preflop: 6.95, Hands: 1,009)
BB: 183.5 BB (VPIP: 28.57, PFR: 10.71, 3Bet Preflop: 0.00, Hands: 28)
UTG: 51.1 BB (VPIP: 22.22, PFR: 0.00, 3Bet Preflop: 0.00, Hands: 9)
MP: 38.3 BB (VPIP: 35.90, PFR: 17.09, 3Bet Preflop: 6.98, Hands: 240)
CO: 36.8 BB (VPIP: 28.57, PFR: 28.57, 3Bet Preflop: 16.67, Hands: 35)
Hero (BTN): 213.9 BB

SB posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has 4 A

fold, fold, fold, Hero raises to 2 BB, fold, BB calls 1 BB

Flop: (4.5 BB, 2 players) T 6 8
BB checks, Hero bets 2.9 BB, BB calls 2.9 BB

Turn: (10.3 BB, 2 players) Q
BB checks, Hero bets 6.6 BB, BB calls 6.6 BB

River: (23.5 BB, 2 players) Q
BB bets 12 BB, Hero calls 12 BB

BB shows 8 A (Two Pair, Queens and Eights)
(Pre 67%, Flop 98%, Turn 100%)
Hero mucks 4 A (One Pair, Queens)
(Pre 33%, Flop 2%, Turn 0%)
BB wins 44.9 BB

What is your selection criteria for cbetting?

My equity and ther fold to cb.

So which of those was the deciding factor to cbet in this hand?

You have 0 equity on this board and this board smacks his range. Just check fold not a good cbet there. Also don't min raise pre.

This ^^^^^^^

only 28 hands not enough to find a good FCB. This flop just crushes his range too much. you have a ton of backdoors and stuff to barrel with here so feel fine just x/f.