You need to determine what kind of GDO it is. Does it have code setting switches? Or is it a rolling code type?
If neither of these, then it is an old analog type, and you will need to replace it. Those things would operate when aircraft flying over used their radio!
A TV remote cannot operate it because they use infrared signaling, not radio signals. It could be a ceiling fan remote, but unlikely. Might be a local radio ham operator.
I suspect you will end up having to replace it. If you do, I recommend a Genie screw drive. It will have the rolling code control and a couple safety features yours may not have. You can get one at a home center for about $155 and install it yourself.