Mix cream cheese with 2 cups salsa and 2 chopped green onions in a bowl until thoroughly combined; lightly stir 2 cups shredded Cheddar and 2 cups Monterey Jack cheese into cream cheese mixture. Spoon a scant 1/2 cup of cheese filling in a line down the center of each tortilla and roll tortilla into a log around the filling. Repeat with remaining tortillas and cheese filling. Place rolled tortillas into a 9x13-inch baking dish.
Spread top of enchiladas with 1 more cup salsa. Lightly mix 1/2 cup Cheddar cheese and 1/2 cup Monterey Jack cheese in a bowl and spread the Cheddar-Monterey Jack cheese mixture over the salsa. Sprinkle with 2 chopped green onions. Cover dish with aluminum foil.
Bake in preheated oven until cheese is melted and bubbly, about 20 minutes.