You need to earn the right to buy them readily
Quick Balls might be my favorite type of Pokeball in the Pokemon franchise. I’d have to really look at everything as a whole, but off the top of my head, endgame collecting with that “will it get captured on the first go?” Quick Ball rush is nigh unbeatable.
That ball type returns in Pokemon Scarlet and Violet, and there’s several ways to pick them up.
Beat six gyms and buy them at the store
As is customary, the more gyms you beat, the more items will unlock at Pokecenter stores all across the game world (and Delibird Presents locations). To unlock Quick Balls, you need to beat six gyms.
This is sort of antithetical to the whole “go wherever you want” concept, as beating gyms will unlock the ability to use higher level Pokemon properly, and provide extra useful items: so progress how you want, but maybe err on the side of gyms first.
Find them in the overworld
You can unreliably find Quick Balls when roaming around the game world, via on-the-ground-pickups.
After hours of exploring, check your inventory: you might have a sizeable amount on hand!
Earn them as Pokedex rewards
Don’t forget: the Pokedex has a poorly explained bonus feature that grants new rewards for every 10 Pokedex entries you acquire. These may provide bonuses for different types of balls: Quick Balls included.