Preheat broiler. Place peppers whole under the broiler and cook until soft and just starting to char, rotating as needed, for 1 to 2 minutes. Remove, let cool slightly, then slice in half from top to bottom. Lay pepper halves flat in a roasting pan.
Preheat/lower oven to 425 degrees.
In a bowl, combine quinoa, shredded squash, basil, tomato sauce, garlic, salt, pepper, and half the cheese.
Divide the quinoa mixture among the four pepper halves. Top with remaining cheese. Bake for 12 to 15 minutes, or until cheese has melted and is starting to lightly brown.