Heat water in a small saucepan over high heat until hot but not boiling; reduce heat to medium-low. Whisk in flour until thickened. Add milk; whisk until flour mixture is thick.
Combine sweet potatoes, green onions, pesto, quinoa, mushrooms, salt, and pepper in a large bowl. Pour in flour mixture; stir until well combined. Transfer to a 9x13-inch casserole dish; top with Cheddar cheese.
Bake in the preheated oven until bubbly and sweet potatoes are cooked through, about 50 minutes.