Can spiriva cause this increased coughing?

Possibly..: Any inhaled medicine can induce coughing as a side-effect, including spiriva. If the cough only occurs right after using the inhaler and is not bothersome, the inhaler can be continued. However, if the cough is bothersome, the inhaler can be switched to an alternate inhaler until one is found that you tolerate. See your doctor so this can be evaluated and a different inhaler tried, if necessary.

Potentially: Any of the inhalers can potentially cause a cough. Often if someone is using an inhaler and it causes cough, it's either because it's the powder form or it's the steroid ingredient. Maybe switching meds will help.