Dice 2 cups cooked chicken into small bite-sized pieces.
In a medium bowl, combine the diced chicken and 1 cup barbecue sauce. Stir until the chicken is evenly coated.
Heat a large skillet or griddle over medium high heat. Place 1 large flour tortilla on a flat surface. Add approximately ? cup of the barbecue chicken mixture to one half of the tortilla.
Top with about 2 tablespoons diced green onions. Sprinkle approximately ? cup shredded Monterey Jack cheese and ? cup shredded cheddar cheese over the chicken mixture. Fold the tortilla in half to form a half-moon shape.
Place the quesadilla on the heated skillet and cook for 2–4 minutes until lightly browned.
Carefully flip and cook for an additional 2–4 minutes until the second side is browned and the cheese is melted.
Repeat the process with the remaining 5 tortillas, chicken mixture, green onions, Monterey Jack cheese, and cheddar cheese.
Let the quesadillas cool for 5 minutes, then slice and serve. Enjoy!