Can I connect a button on my motorcycle to the main button on a garage door remote?
If the garage door system is using the same authentication method as I've seen used in cars locks, you have to watch out with using something like indicators. 

The lock and the remote use the same number-generator to generate a new random number every time you use it. The lock will accept any of the next 255 or so numbers. This way, it will still work after a few failed transmissions. Once the lock receives any of the allowed next 255 numbers, it will update it's internal counter to "catch up" with the remote. Because there are so many possible numbers, the chance of someone else transmitting one of the allowed numbers is tiny (on the order of meteor hitting your head).

However, if you bind the button to your indicators, every time you use it the remote will jump to the next number. It's very conceivable that after a decent drive, you used the indicator more than 255 times - and now your remote will not open the garage door anymore.

That said, why do you want to mount it to an existing button? Would it not be easier to somehow remove the button in the current remote, stick it somewhere where it's nice and tucked away (but not surrounded by metal), and then run a cable from there to your controls and add a small button?