No… don’t do this. I’ve explained this elsewhere, but basically you are going to burn shit out.
A button is a simple device. It shorts two contacts. That’s all. It’s literally just a piece of metal that you physically press down that shorts two conductors together. Typically speaking, the circuit designed to detect user interface button presses (basically all low voltage digital devices) has it’s own ideas about the electrons it wants to push down one of those conductors, and detect coming back in on the other conductor once the button is pressed (shorting them together).
Think of it like a toy train set… a big circle… and you’re there, cross-legged on Christmas morning, playing with your toy train set… and you have a little drawbridge in part of the track so imaginary sailboats can make their way through… and your drawbridge is up… so your train waits… well that drawbridge is the button. Once the button is pressed, the drawbridge is lowered and your little train of electrons can mozey on over the drawbridge and continue on their way.
Now let’s say your dad comes in with a hotwheels toy car track and glues it to the drawbridge and says “here come the cars motherfucker!” and instead of a tiny toy car making it’s way down the hotwheels track, he drives his 1967 Ford Mustang GT over you, your shitty little train track, the Christmas tree, and the dog. That’s what it’ll be like if you try to connect the access controller’s output wire directly to the button of your garage door remote.
So what’s the answer? A relay. A relay is what? It’s a device that shorts conductors together on command. Sounds kinda like a button right? That’s right, it is just like a button… only a button that you can control with another circuit! You connect a relay’s electromagnetic coil pins to the white output wire and GND of the access controller… when the output wire goes active, the relay’s coil will physically pull the contacts closed. Therefore, you solder the relay’s contact pins (the common pin and NO or “normally open” pin) to your garage door opener’s button conductors, and boom… you have an alternative, access controller controlled way to short the conductors of the button in your garage door remote… and you, the dog, your shittly little train track, and the Christmas tree all get to live (well, not the Chistmas tree, it’s already dead… just doesn’t know it yet).