Bring water and quinoa to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until quinoa is tender and water has been absorbed, 15 to 20 minutes.
While quinoa cooks, mix black beans, red bell pepper, spinach, barbeque sauce, garlic, cumin, and cayenne pepper together in a saucepan over medium heat. Stir in Cheddar cheese and cook, stirring occasionally, until the cheese is completely melted, 5 to 10 minutes. Remove from heat and stir in cooked quinoa.
Place tortillas on a flat work surface. Spoon filling in a line across the middle of each tortilla; fold in opposing edges to overlap the filling. Roll up as you would burritos.