You've run across a problem of words. Most likely all you need to do is call the doctor's office and ask the billing staff to rebill it under a different code. Nonspecific neck pain means there's no underlying disease such as cancer, arthritis, fibromyalgia, and so on.
The cause is more likely to be linked to the way the muscles contract or the joints line up. Perhaps a past injury resulted in a tear in the ligaments. The neck may lack stability. The muscles try to compensate. They get overworked and tired and start hurting.
We call this a biomechanical problem. That's just one example of how changes in the normal biomechanics can lead to pain and problems. Calling it "nonspecific" just says the cause is unknown, but not from a systemic illness.