A common automation project is to control a garage door using some sort of a relay output from an alarm or automation system using Zwave, Insteon, UPB or WiFi IO module. This is accomplished by connecting the normally open relay terminals of the IO module, across the two garage door control wire terminals.
To test if this will work, use a small piece of wire and short it briefly across the two terminals that go to the garage wall button. If that opens/stops/closes the door and leaving it shorted for a couple seconds doesn't cause any other issues then you can most likely control your garage door opener in this manner.
Note that legally in many jurisdictions there are implications with remotely closing a garage door with no additional safety precautions. Check your local codes or bylaws. There is a Zwave garage door controller from Linear that allows control of a garage door but offers a visual and audible warning first, which creates a safer environment and satisfies most legal requirements.