Can I connect my generator transfer switch to a subpanel instead of to the main panel?
You could simply pull a properly sized cable from the transfer switch, out to the garage. Then install a generator input receptacle in the garage.



There's likely a knockout in the bottom of the transfer switch to allow for this. If not, you may be able to add your own (check with the manufacturer).  

Moving the transfer switch, and migrating circuits is is likely going to be a huge headache.  It's much simpler to pull a single new cable.