Can I use a transistor to control my garage door?
It may be possible to use a MOSFET but a lot would depend on the nature of the signalling that the goes on the wire between the opener and the wall mounted control unit. If the control unit is some type of smart unit then that may limit your options. In the end we cannot guess at all whether any particular MOSFET would work and how it would need to be wired in.

I found it much more convenient to use a low cost eBay wireless relay module and wire the NO and COM terminals of the relay directly across the push button switch contact connections. The wireless module was controlled by a carry around fob with a push button that operated at 315 MHz. You can also get low cost 315 MHz wireless transmit modules on eBay that can easily be connected to a microcontroller that mimics the protocol of the fob in software.