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I am presuming that neither the old or new version is a Starter Edition.
For full versions (such as Basic and Deluxe) the 2013 version will be able to open and convert the 2006 files without loss of information.
Installing Quicken 2013 on the computer that already has any Quicken 2004 or newer will install and automatically convert your Quicken data files.There will also automatically be a copy of the prior Quicken data file in a folder on your hard drive.
Be aware that the password XXXXX 2013 has to be fourteen characters or less so change it before converting if it is longer than fourteen characters.
More information is available at the support article at http://quicken.intuit.com/support/help/install--register--and-convert/convert-data-from-an-earlier-version-of-quicken/GEN82390.html
Please ask if you need clarification.