See ENT or Chest Physician. Allergic rhinitis or Sinusitis needs to be excluded. A CT scan of Sinuses and throat may be useful. CT scan of the chest may identify early lung, pleural lesions not seen in normal chest x-ray. Spirometry (lung function test), Hitamine challange tests, and Allergen skin/blood test, may also help to exclude allergic cough. Cough related to HIV infection would have other symptoms (like loss of appetite, weight loss, fever etc) and CXR would be abnormal.
It could be from smoking and your lungs are trying to slough off the old cells. Do you have any symptoms of TB other than the cough, such as night sweats, extreme fatigue etc.? You can easily get a TB skin test (unless you got the bCG vaccine) or Xray for that if you are worried about. I doubt your test for HIV was wrong. If you were exposed did you get tested 3 months afterwards? If not you could get that done to reassure yourself but I don't think just having a cough would be a sign of HIV. You have other symptoms. As the other post said it might be good to also see an ENT doctor to check your throat and see if something is making you cough.
How about seeing an ENT doctor!