Could it happen that a nut allergy cause a sinus infection?

Unlikely: One does not lead to another.

Yes: Sinusitis may result from allergic reactions which involve the upper respiratory tract. By creating swelling of the tissues inside the nose you can close off drainage from the paranasal sinuses and the subsequent accumulation of fluid can secondarily result in sinus infection. This is one of the significant predisposing factors to sinusitis.