Firefox stores your bookmarks in a database file
places.sqlite under your profile
folder. Because the database filename appears to be hardcoded, it's probably not designed to be moved to a different location.
Instead of trying to share just your bookamrks, why not share your entire Firefox profile between Ubuntu and Windows? Leave your profile folder on Windows, then update profiles.ini on Ubuntu to point to your Firefox profile folder under Windows. I've had good experience transferring Mozilla profiles across OSes, although YMMV. Whatever you do, be sure to make a backup copy of your profile folders before you do anything.
Google shows that some people have had success with this setup. See this HowTo, for example. It looks like certain extensions might have issues doing this, though.