Warm a nonstick skillet over medium heat. Lightly butter one side of the tortilla and place it in the hot pan. Sprinkle half the cheese on one half of the tortilla and top with the turkey and the green onions. Sprinkle the remaining cheese over the chicken.
Fold the tortilla over on itself and let it cook for 2-3 minutes. When the cheese has begun to melt and the bottom tortilla is browning slightly, flip the folded quesadilla over to cook the other side.
Continue to cook until all the cheese is melted and the tortilla is crisp around the edges. Using a spatula, transfer the quesadilla back to a cutting board and cut it into wedges while it is hot. Serve warm.