Can sinus allergies cause coughing ?

Absolutely!: Sinus fluid often drains down the back of your throat, causing a cough reflex. This is fairly common. If you have known allergies and aren't having fever or green/bloody drainage, it's really not a problem.

Yes: The same allergens that trigger your sinuses to fill can also cause inflammation in your nose and cause the secretions (mucous) to drip from your nose into your throat causing a pretty significant, at times, cough.