In a medium bowl, toss chicken pieces with fajita seasoning and 2 tablespoons olive oil using tongs. Set aside.
Heat 2 tablespoons olive oil in a 10 to 12 inch nonstick skillet over medium-high heat.
Sauté the seasoned chicken until cooked through and no longer pink, about 6 to 8 minutes. Transfer to a heatproof bowl.
In the same skillet, add diced bell peppers and onion; cook until onion is opaque and peppers are softened.
Return cooked chicken to the skillet, stir to combine and warm. Transfer back to the heatproof bowl.
Reduce heat to medium and add remaining 1 tablespoon olive oil to the skillet.
Place one tortilla in the skillet. Sprinkle ¾ cup shredded cheese over half the tortilla.
Top cheese with ¼ of the chicken and vegetable mixture; fold the tortilla over to cover filling.
Cook quesadilla until golden brown on the bottom, then flip and brown other side, about 1 to 3 minutes per side.
Repeat with remaining tortillas and filling.
Transfer quesadillas to a cutting board and slice each into 3 triangles. Serve hot.