Heat a large nonstick skillet coated with cooking spray over medium-high heat. Add ham, green chiles, and mushrooms; sauté 5 minutes. Remove from pan. Wipe pan clean.
Place 1 tortilla in pan. Sprinkle 2 tablespoons cheese over half of tortilla; arrange 1/2 cup ham mixture over cheese. Sprinkle with 2 tablespoons cheese; fold in half. Cook 1 minute on each side or until cheese melts. Repeat procedure with remaining tortillas, cheese, and ham mixture.