It could be asthma or it could be from an annoying post nasal drip. I would have a doctor check for allergies, irritants, and chemical sensivities. I would also have his sinuses checked for blockages and his nasal passages checked for inflammation. Is he doing nasal rinses or if he is already, can he do them more often? Ask the doctor about nasal sprays.
I know for me, if I go into a coughing fit, it is an asthma reaction and I have to take a couple of puffs of a bronchodilator. (But please check with a doctor before he does this) Does he notice any triggers before a coughing fit like cold air, or cold drink, or going into a store, or eating somnething?
Has he even been tested for asthma? And is he on the typical CF drugs which include a lot of asthma meds?