Mix the shredded chicken and barbecue sauce thoroughly in a bowl until the chicken is evenly coated.
Warm 1 tablespoon of olive oil in a skillet over medium heat.
Lay one tortilla flat in the skillet, then sprinkle 1/2 cup of shredded cheese on it, spread half of the BBQ chicken mixture evenly over the cheese, and add another 1/2 cup of cheese on top.
Place a second tortilla on top and cook for about 3 to 4 minutes until the bottom tortilla turns golden and the cheese starts melting.
Flip the quesadilla carefully and cook for an additional 3 to 4 minutes until the other side is golden and the cheese has melted completely.
Transfer the cooked quesadilla off the skillet and repeat these steps for the remaining tortillas and BBQ chicken filling.
Cut each quesadilla into wedges, garnish with chopped green onions if you like, and serve immediately while warm.