It is highly likely that you have chronic
bronchitisAcute bronchitis
Bronchitis
Bronchitis and normal condition in tertiary bronchus
Cause of acute bronchitis
Cause of chronic bronchitis
Chronic bronchitis
Croup
Industrial bronchitis or chronic
obstructiveAcute bilateral obstructive uropathy
Obstructive uropathy pulmonary disease (COPD), which is a combination of chronic bronchitis and emphysema. It may be that your cough reflex was suppressed by the cigarette smoke when you were smoking and with smoking cessation it has regenerated to clear the excess mucus from your lungs.
However, you should not assume that that is the problem. It may be that the cause of your cough is unrelated to smoking cessation. Possibly it is caused by another lung disease. For this reason, you should see a pulmonologist, also called a lung specialist, to determine the actual cause of your cough and initiate treatment for it.