Have you visited any general physician because it could be due to some allergies? until there hasn't done any diagnosis from a physician it's difficult to say anything, also as you have mentioned that your cough starts when you are sitting idle or when you are free? The point is when you are free you need to give it thought what are you thinking at that point of time as mind is never idle or free from any thoughts, so keep a track on your thoughts if they are anxious, stressed.... than probably there is a chance of somatic cough syndrome but it's not a diagnosis yet. For more help you can sure book your session with me to have more details.
Next Steps
track your thoughts and see if the timings when you are free what sort of thoughts are running into your mind?
track your lifestyle if it's healthy enough?
track your diet ?
Health Tips
Start with deep breathing exercise that will help calm down your present symptom if it is related to emotional health concern.