In a medium sauce pan heat the olive oil over medium-high heat. Add the onions and garlic. Cook about 5 minutes or until the onions are translucent.
Add the mushrooms, thyme, salt and pepper and cook for about 7 minutes more.
On a work surface, spread 4 of the tortillas. Sprinkle the Monterey jack cheese over the bottom of each tortilla. Spoon evenly the mushroom mixture on top of the cheese, then top with provolone and mozzarella cheese. Top with another tortilla.
Heat a large nonstick skillet with cooking spray over medium heat. Carefully place 1 quesadilla in pan and cook 2 minutes. Using a spatula, gently flip the quesadilla and cook an additional 2 minutes until lightly browned and cheese is melted. Repeat with second quesadilla. Serve warm.