Tranquill should get STAB from both Normal and Flying moves, right? So why is a base 40 Quick Attack doing the same damage to enemy mons as a base 60 Air Cutter? It's as if Tranquill is not getting STAB on her Air Cutter or something.
Quick Attack is a physical move, while Air Cutter is a special move.
The damage of physical moves is calculated based on the user's Attack stat and the target's Defense stat, while the damage of special moves is calculated based on the user's Special Attack stat and the target's Special Defense stat.
You can recognize the type of moves by the icon in their description: for physical moves, and for special moves.
Tranquill's Attack is much higher than its Special Attack, so Quick Attack will deal more damage than Air Cutter against a target with equal defensive stats. Physical attacks like Wing Attack and Aerial Ace are better on a Pokémon with higher Attack like Tranquill.