That sounds like it's a server config problem that is most likely a result of limitations GoDaddy has set on their server.
You might be able to get resolution via their tech support team (ask them to address the charset issues in their mysql configuration).
However, since your database sits on a server that is separate from your hosting server, you may have some limitations in using the server-side tool to do the backup.
You'll likely have to use phpMyAdmin instead, or some backup option offered by GoDaddy's control panel, if any.