In a medium bowl, combine the shredded chicken and BBQ sauce. Mix well until the chicken is fully coated.
Heat a large skillet over medium heat and add the olive oil.
Place one tortilla in the skillet and sprinkle half of the cheddar and mozzarella cheese evenly over the tortilla.
Spread half of the BBQ chicken mixture over the cheese, then sprinkle with half of the green onions if using.
Top with another tortilla and cook for about 3-4 minutes, or until the bottom tortilla is golden brown and the cheese begins to melt.
Carefully flip the quesadilla using a large spatula and cook for an additional 3-4 minutes on the other side until golden brown and the cheese is fully melted.
Remove from the skillet and let it cool for a minute before cutting into wedges. Repeat with the remaining tortillas and filling.
Serve warm with sour cream and salsa on the side.