Hi!
Well, Kudos to you for quitting smoking! Well done! It is natural to experience some cough and there may be brown pieces in it due to gastric reflux.
Treatment is a combination of drugs to reduce the acid and lifestyle changes. You will need to take a combination of medications (under medical supervision) like a proton pump inhibitor such as omeprazole, lansoprazole or pantoprazole empty stomach in the morning and an antacid gel after meals for complete relief. Possibility of H pylori infection too should be looked into by a carbon urea breath test and a combination antibiotic tried. Many a times a persisting H pylori infection can be the cause behind acidity not responding to treatment.
Life style changes that will help include: Drink cold milk. Avoid heavy meals and eat frequent small meals. Avoid too much of caffeine, tea, fried food and drinks both alcoholic and non alcoholic fuzzy ones. Avoid heavy exercises within 4 hours of a heavy meal. Raise the head end of the bed by pillows to 30 degrees. Avoid lying down for least 2 hours after food. Maybe these tips will help you.
Do discuss with your treating doctor. Take care!