Preheat oven to 425 degrees F (220 degrees C). Line a rimmed baking sheet with parchment paper.
Place cooked chicken cubes in a food processor. Pulse until chicken resembles crumbs. Add shredded Cheddar and Monterey Jack cheese and jalapeno cream cheese. Pulse 2 or 3 times until mixed. If mixture appears dry, add more cream cheese, 1 tablespoon at a time.
Place a heaping teaspoon of chicken-cheese mixture on the bottom corner of a wonton wrapper. Leave a 1/2 inch border around wonton. Brush the edges with water and fold over opposite corner to form a triangle. Press to seal edges. Repeat for all wontons.
Place wontons on prepared baking sheet. Spray lightly with cooking spray. Turn; lightly spray other side of wontons with cooking spray.
Bake in preheated oven until the edges turn brown, 8 or 9 minutes.