Thanks for your question on Health Care Magic..
No this is not normal after quitting cigarettes.
You might be having bronchitis
Bronchitis is inflammation of airways and it is common in smokers.
Cigarette smoke causes constant irritation and inflammation of airways.
And it causes bronchitis and coughing.
So better to get done PFT (Pulmonary Function Test) for the diagnosis of bronchitis.
It will also tell you about severity of the disease and treatment of bronchitis is based on severity only.
You may need inhaled bronchodilators and inhaled corticosteroid.
So better to first diagnose yourself and then start appropriate treatment.
Hope I have solved your query.
Wish you good health. Thanks.