In a mixing bowl, combine the shredded chicken with BBQ sauce until well coated. Set aside.
Heat olive oil in a large skillet over medium heat. Add the sliced bell pepper and red onion, sautéing until they are softened, about 5-7 minutes.
On a clean surface, lay out one tortilla. Spread a generous amount of the BBQ chicken mixture on half of the tortilla. Top with a handful of shredded cheese, then add some sautéed bell pepper and onion.
Fold the tortilla over to create a half-moon shape.
In the skillet, place the folded quesadilla and cook for 3-4 minutes on one side until golden brown and crispy. Carefully flip it using a spatula and repeat on the other side for an additional 3-4 minutes.
Once both sides are nicely browned and the cheese has melted, remove the quesadilla from the skillet and let it sit for a minute before cutting it into wedges.
Repeat the process with the remaining tortillas and filling.
Garnish the quesadilla melts with fresh chopped cilantro and serve with sour cream on the side for dipping.