Is it possible that the mercury in fillings can cause sinus problems?

No. Sinus disease is the result of the impairment of normal drainage and airflow through the sinuses. The drainage openings can get obstructed by viral infection, allergy, septal deviation, nasal polyps, tumors, barotrauma among others. Mercury fillings play no role in acute or chronic sinus disease.
Sinus issues. I have never seen any issues with dental fillings causing sinusitis unless there was some rare complication where the filling material got into the sinus and caused sinusitis. This is very rare. I would see an ent. If your sinus issues persist.