Can I still build muscle if only eating carbs and very little protein, while still limiting my calorie intake?

Not likely. A balanced diet is key with recommended amounts of protein, healthy fats, and carbohydrates. Given you described diet it is possible to lose weight, yet increase your percent body fat and slowdown metabolism all of which are unhealthy. You also want to eat a diet that offers variety, optimizes your health and is appropriate for you lifestyle. Fad diets are not the answer.