Having not installed Office 2013 (too buggy for my liking - I'm staying well away from it, as are various other MVPs), I can't be 100% certain about the process.
What I do know is that the free upgrade, which is what I assume you're referring to, was only for Office 2010 users who bought & installed their software during a certain period. You may need to reinstall Office 2010 before installing Office 2013 as I can't see any reference to the kind of upgrade version MS used to produce. Even then, I can't be sure it'll work as MS hasn't published anything on this aspect. Conversely, you may only need to supply the Office 2010 product key when installing Office 2013. Alternatively, you might use a product like The Magical Jelly Bean Keyfinder (
http://www.magicaljellybean.com/keyfinder/) to retrieve the Office 2013 product key and, if it differs from the Office 2010, use that for the new installation.