Heat a grill pan over medium-high heat. Add the peppers and cook for 5 minutes, stirring occasionally. Drizzle with the olive oil and season with the chili powder and garlic powder. Cook for another 5 minutes, stirring occasionally.
Lay out two of the tortillas on a work surface. Spread each with 2 tablespoons of salsa verde and top with half of the cheese and half of the grilled peppers. Top each with one of the remaining tortillas.
Heat a large non-stick skillet over medium heat. Place one of the quesadillas in the skillet and cook for 3 minutes per side or until golden brown and crispy.