In a medium bowl, combine the shredded chicken and BBQ sauce. Mix well until the chicken is fully coated.
In another bowl, mix the cheddar cheese, mozzarella cheese, garlic powder, onion powder, salt, and black pepper.
Heat a large skillet over medium heat and add 1/2 tablespoon of olive oil.
Place one tortilla in the skillet and sprinkle about 1/4 cup of the cheese mixture evenly over the tortilla.
Top the cheese with 1/2 cup of the BBQ chicken mixture, then add another 1/4 cup of the cheese mixture on top.
Place a second tortilla on top and press down gently. Cook for about 3-4 minutes until the bottom tortilla is golden brown and the cheese starts to melt.
Carefully flip the quesadilla using a spatula and add the remaining olive oil to the skillet. Cook for another 3-4 minutes until the other side is golden brown and the cheese is fully melted.
Remove from the skillet and let it cool for a minute before cutting into wedges. Serve with sour cream and sliced green onions if desired.