The easiest way to transfer just your bookmarks is probably by exporting them to HTML. See:

• Export Firefox bookmarks to an HTML file to back up or transfer bookmarks
• Import Bookmarks from an HTML file

Other than that, you can transfer the places.sqlite file from your profile folder. See Back up and restore information in Firefox profiles.

Why not just open the browser on the new computer, sign into the same Firefox account and watch the magic of Sync? As long the options are checked off in Sync settings on your old computer. see screenshot