Coughing is a common condition. It can be associated with hyperventilation. That being said, if you are having coughing fits severe enough to cause hyperventilation you should likely see your primary care doctor
to get to the bottom of your cough's cause.
Coughing is a normal reflex. The body has developed the reflex to clear the airways. What happens in the cough is that firstly the airway closes, in order to increase the pressure in the airways. Once the airway has reached a certain pressure, the airway opens and the built up pressure helps clear the airways upward. Therefore the cough actually decreases ventilation.
That being said, when some people cough many times in a row their ventilation decreases significantly over that period. The body senses this decreased ventilation (through a transient increase in the carbon dioxide in the blood) and therefore can cause a overcompensation by hyperventilating. The body tries to blow off the built up carbon dioxide and can overshoot. This can give the symptoms of hyperventilation.
This overshoot will only occur in severe coughing fits. The real question is what is causing this. Do you have an infection? Do you have another cause of chronic cough?
I encourage you to see your doctor to have this evaluated if it continues.