Coat a large skillet or griddle with cooking spray. Set pan over medium heat to preheat. Using a fork, pull chicken meat from bone in shreds.
Arrange tortillas on a flat surface. Top 1 side of each tortilla with an equal amount of chicken, mushrooms, red peppers, and artichokes. Top vegetables with shredded cheese. Fold over tortillas, making a half moon. Place quesadillas in (or on) prepared pan and cook 3 minutes per side, until golden brown and cheese melts.