If I use Quick Attack in Pokemon Legends Arceus, and I activate strong style, do they cancel each other out, or does strong style override?
Quick Attack, when used as a Strong Style move, adds 2 to the user's action speed. This doesn't mean much on its own — you have to look at it in the context of the user's and the target's initial action speed stats. Generally this means the opponent will have a few consecutive turns as a result of you increasing your own action speed.
The main takeaway is that Quick Attack is not a priority move in Legends: Arceus. There is nothing to "cancel out".
Action speed (Bulbapedia) Quick Attack (Bulbapedia)