Let me first start with a security disclaimer. It is not recommended to use ST in order to automait your garage doors as there were and still are issues with garage doors triggering while people are away or sleeping. That said, I have automated my garage door and have posted a project on how to some time ago. Please read it completely and I stess the part about failsafe: Install a Smart Power outlet in line with your garage door opener before you automait in order to be able to secure your garage door.
If the button for your garage door is a simply contact switch then you can install the GD00Z-4 zwave garage door controller. (The button you listed is just a basic contact so this device should work with your garage.)