No, there shouldn't be any problems. The only thing you will need to do is make some changes to the configuration.php so that the database setting match your database on the GoDaddy account.
You may however some some issues with some extensions. In an ideal world, if the extensions you're using have been developed using the most up to date coding standards, then there will only be some deprecated code, which should not affect functionality. However, I've seen lots of extensions for Joomla 2.5, that use Joomla 1.5 coding standard and this is what will cause issues. It does occur sometimes with big complex extensions such as jDownloads, and they do not provide a stable Joomla 3.x compatible version, thus you would have to find an alternative.
Due to this, be sure to keep a backup of your Joomla 2.5 site and I would suggest migrating on a test server, not the live site, to ensure that everything works smoothly.