While I hate to make wide sweeping generic statements about all garage door openers, every single unit I have ever seen or installed has a set of contacts somewhere on the opener that allow you to connect a normally open push button switch. This is probably the same set of contacts that go to your button in the garage to open or close the door. Close the circuit with a momentary switch and the door either opens or closes depending on its current state. In most cases the wiring for the eye at the lower part of the door connects to one or both of these or has a separate set of connectors you really don't need to worry about.
Simply connect a relay output on the M1 (Normally Open) to the same two terminals and pulse the relay for one second to intitiate movement. If you want to get fancy you can use a set of sensors to determine if the door is up or down. This way if you want to automatically close your door at a specific time you read the value of the sensor (a simple switch, probably magnetic) and determine whether or not you need to pulse the relay to close the door.
A garage door is probably the easiest device to automate in an entire house but you must be careful moving the door without some sort of warning if you do it at times when people might be around the garage or the door.