Combine onion and vinegar in a bowl; let marinade for 5 minutes. Drain, reserving the vinegar for another use, such as salad dressing.
Warm 2 tortillas in a non-stick skillet over medium-high heat for about 45 seconds, then flip. Pull the tortillas up the edges of the pan so they are no longer overlapping.
Working on one half of each tortilla, sprinkle one-fourth of the cheese, cover with roasted turkey and top with one-fourth of the onion.
Fold the tortillas in half, flatten gently with a spatula and cook until the cheese starts to melt, about 2 minutes. Flip and cook until the second side is golden, 1-2 minutes more.
Transfer to a plate and cover to keep warm. Make 2 more quesadillas with the remaining ingredients.
