Dice the chicken into smaller pieces. Stir together the chicken and the barbecue sauce into the chicken is thoroughly coated with the sauce.
Heat a large skillet or griddle over medium high heat. Place down one flour tortilla. Place the chicken mixture, green onions and each of the two types of cheese on one side of the flour tortilla. Fold the tortilla over to make a half moon shape.
Cook for 2-4 minutes on each side until the tortillas are browned slightly. Continue doing this process until all of the quesadillas are made.
Let the quesadillas cool for 5 minutes, then slice and serve. Enjoy!