I don't know where you can find out for sure about the belt containing asbestos. Maybe call the manufacture. I read about this in Dr. Clark's book a Cure for All Diseases. Since that time I think about it every time I go into the laundry room. I turn the fan on and shut the door as a precaution when drying cloths. Line drying cloths is better all the way around if you live in the country.
Something else to think about, the book was written in the 1990's. Maybe the new belts are asbestos free, but I would not count on it.