Depending on the voltage required to trigger the relay for the gate opener you could setup an analog port or even connect it to the page relay on the phone system. It's 24V off the phone system if I am not mistaken on the page relay port, and analog ports are definitely 24V.
You could setup your menu to dial the analog port, not sure if you can program the menu to trigger the page relay but you may be able to. Paging is usually accessed by entering a feature code on the Norstar. You may also be able to connect it like a doorphone, I have seen those used to trigger security door releases but I have not done one so I don't have specific instructions