I recall looking at a wiring diagram for your garage door opener-motor-unit and the 4 terminals in your picture were just for the limit switches that stop the door travel. You can trace the wires to these limit switches and I expect they do not go to your wall-mounted controller.
The “learn button” is to pair the wireless wall-mounted controllers to the opener-motor-unit. Even if there is “wire” to your wall-mounted controller, it may just be 24vac power for the controller, but the actual signal to open/close is sent wirelessly. So I expect that you don’t have a simple wall-mounted wired switch, which is what the Gocontrol Zwave device emulates. You can clarify your wall-mounted controller by taking a picture of it on the wall, and take off the cover and show its connections.
The workaround is to solder the Gocontrol output to the mechanical contacts of the button in the wireless wall-mounted controller. There are several instances of forum posts doing this. I’ll see if I can find one with this crappy forum search: