Both a thyroid nodule and a chronic cough are concerning symptoms. I would recommend seeing your primary care physician
to deal with both.
A thyroid nodule could cause a cough--although this would be rare. Essentially, the thyroid is a gland in the neck that produce a hormone for energy metabolism. A thyroid nodule can form from a growth--either benign or cancerous. In general, the risk of cancer is high enough that we often recommend almost every nodule is biopsied. I would see your primary doctor
who can do an ultrasound
to see if there is a nodule and then do a simple biopsy.
As for your question if it can cause cough--there are two potential ways. Again, both are rare. One would be if the nodule is compressing on the air pipe. The cough is the protective reflex--therefore a cough to reopen the airway would be expected. Second, the recurrent laryngeal nerve is near the thyroid although lower. If the nodule were to compress the nerve this could "tickle" the throat causing a cough.
As they are rare--they are most likely unrelated. Other causes of chronic cough include smoking, medicines, heartburn, asthma and allergies. These can be treated.
I would recommend seeing your primary doctor. While it is possible they are related, it is unlikely. Regardless, both the thyroid nodule and the chronic cough need to be addressed. Good luck!