In a large bowl, combine the shredded chicken, mozzarella cheese, cheddar cheese, cream cheese, taco seasoning, diced bell peppers, and diced onion. Mix well until all ingredients are evenly combined.
Heat 1 tablespoon of olive oil in a large non-stick skillet over medium heat.
Place one tortilla in the skillet and evenly spread half of the chicken and cheese mixture over one half of the tortilla. Fold the other half of the tortilla over the filling.
Cook for about 3-4 minutes on each side, or until the tortilla is golden brown and crispy, and the cheese is melted.
Remove the quesadilla from the skillet and cut it into wedges. Repeat the process with the remaining tortillas and filling.
Serve warm with salsa and sour cream on the side for dipping.