Could a sinus headache cause ringing of the ear?

Possibly: Ear ringing is a poorly understood symptom and transient ear ringing can occur with jaw pain, neck pain as well as sinus issues.

Sinus headache: A sinus could be associated with tinnitus, or ringing in the ear. However, I am assuming that all the syndromes appeared simultaneously.

Tinnitus: Tinnitus - or ringing in the ears - has a number of different causes. It is not out of the question that a sinus infection could exacerbate tinnitus, but it is not common either. Medications, stress, acoustic trauma, and many other things can trigger tinnitus.