Heat 2 teaspoons of the oil in a large nonstick skillet over medium-high heat. Add the bell pepper and cook, stirring frequently, until tender, 5 to 7 minutes. Set the skillet aside and use again for cooking the quesadillas.
Place the cooked bell pepper in a large bowl. Stir in the cheese, beans, corn, and barbecue sauce. Divide the mixture evenly and spread over half of each tortilla. Fold over, press down gently and set aside.
Heat 1 teaspoon of the remaining oil in the large nonstick skillet over medium-high heat. Add two of the quesadillas and cook, pressing down occasionally with a spatula, until the tortillas become crisp and golden on the bottom, 4 to 5 minutes. Flip the tortillas and cook an additional 3 to 4 minutes.
Repeat with the remaining oil and quesadillas. Cut into triangles and serve.