The reason you have a cough is because toxins are being expelled via the lungs. The body detoxes toxins in a few basic ways: via the skin, via the bowels, and via the lungs. The lungs are the dumping ground of the body, so if you are killing something off such as candida, the toxins will flood the lungs for some people.
I had a very bad cough after starting the diet but it improved within a few weeks. If it becomes a long-term cough and not a short-term cough, I would consult a doctor but I think this is a strong case of cause and effect.