Preheat the grill to high. Mix the chicken, onions, red cabbage, cheddar and carrot together in a bowl.
Place a non-stick frying pan over a medium heat, then place 1 tortilla in the pan and top with the chicken mixture. Place another tortilla on top and press down. Cook for 2-3 minutes until golden on the underside, then remove from the heat and place under the preheated grill.
Cook for a further 2-3 minutes, until the quesadilla is golden on top and the cheese has melted. Transfer the quesadilla to a chopping board and leave to cool for 2 minutes, then cut into triangles for a tasty lunch.