So to see if I have a handle on the problem…
You want an app (or web page, or whatever) that can open a garage door opener device. You do not want to connect it to a wired button. I would surmise that this is possibly because the wired button and opener unit is out of your home wifi range (As it is for my garage.) So you want something to basically open the door from within your home.
I would probably hack a Raspberry Pi to press the button on a remote. A couple of wires to a small relay would do the trick nicely. Most garage door openers run on a fairly limited RF range, and have inexpensive substitutes available. You need only open yours, read the dipswitch code from yours and duplicate it on the second unit. This is probably the most inexpensive route to generating the coded frequency pulse to open the door. Now the question is will you be in wifi range of the Raspberry Pi, or will you need to have Internet access.
If you will be in range of the Pi, then most secure would be to run the Pi as an access point. This way, you are only vulnerable from within a very small geographical area, someone would need to be in range of your house in order to hack into the garage door opener. (Did I hear something about access to an alleyway?) I suppose something similar might work for access to underground parking at an apartment building or condo complex, provided of course that you can open the door from within your unit.
If you will not be in range of the Pi, you will need Internet access. There are a number of ways to accomplish this with relatively secure means, but I’d highly recommend implementing an access log so you can be sure that nobody is opening and closing the gate at random times just because they can.