Bring water and quinoa to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until quinoa is tender, 15 to 20 minutes.
Heat olive oil in a large skillet over medium heat. Add onion and cook until softened, about 5 minutes. Add turkey and season with salt and pepper; cook and stir until turkey is no longer pink, 5 to 7 minutes. Add mushrooms and cook until softened, about 5 minutes.
Preheat the oven to 350 degrees F (175 degrees C).
Transfer cooked turkey mixture to a large bowl. Add cooked quinoa, tomato sauce, and tomato paste. Stir well; add additional tomato sauce as needed, until filling has the consistency of a casserole.
Arrange red bell peppers in a baking dish and spoon an equal amount of filling into each.
Bake in the preheated oven for 45 minutes. Remove from oven and top each stuffed pepper with Cheddar cheese. Continue baking until cheese is melted, about 10 minutes more.