If you have a cough that lasts for months and doesn't go away or the coughing seems to get worse you should see a doctor or go to the emergency room or urgent care center to get checked out just to be safe.
Coughing that lasts for months at a time is usually not normal and can be caused by things such as asthma, allergies, gastroesophageal reflux disease (GERD), or bronchitis.
You could also have a lung condition or lung disease as well so I would highly advise at least seeing your doctor who may then send you to the hospital to be checked out.
You could have a simple bacterial infection that is causing the coughing that lasts for months and taking some antibiotics can help clear up the infection.
You may also have breathed in some irritants that may be affecting your lungs as well so I would go to the doctor at least to get checked out.
If the coughing gets worse go to the emergency room or urgent care center to get checked out just in case it's something more serious.