In a large mixing bowl, combine the shredded chicken, ranch dressing, diced red and green bell peppers, red onion, and half of each type of cheese (cheddar and mozzarella). Mix well until all ingredients are evenly coated. Season with salt and pepper to taste.
Heat 1 tablespoon of olive oil in a large skillet over medium heat.
Place one tortilla in the skillet and evenly spread half of the chicken mixture over one half of the tortilla.
Sprinkle the remaining cheese (both cheddar and mozzarella) over the chicken mixture, then fold the tortilla over to create a half-moon shape.
Cook for 3-4 minutes on one side until the tortilla is golden brown and crispy, then carefully flip and cook for an additional 3-4 minutes until the other side is golden and the cheese is melted.
Repeat with the second tortilla, adding more oil to the skillet if necessary.
Once cooked, remove the quesadillas from the skillet and let them rest for a minute. Cut each quesadilla into wedges.
Serve the quesadillas warm, garnished with fresh cilantro if desired, and with extra ranch dressing on the side for dipping.