Run the database tools within Extensions > Manage > Database and see if it can upgrade the database for you. If that doesn't work yes you can migrate CBs tables by exporting and importing all tables prefixed with _comprofiler, _cbsubs, and _groupjive.
The tricky part is your Joomla users table and their ACL assignments as well as any permission you may have saved. You can try installing CB manually, using CB Juice to export from your old site, import into your new site, then override your CB tables.
There's not really a process in CB or Joomla to do what you're trying to do (partial migration of data) and for the most part is entirely manual. Maybe Akeeba Backup or similar extensions has a tool to help with partial migrations.