No, the Free Trial uses the same client as the full version, so you will not be required to uninstall the Free Trial client.

For more details and the required steps for upgrading from the Free Trial to the full version, please visit http://support.na.square-enix.com/faqarticle.php?id=5382&la=1&kid=70905

For other free trial-related questions and concerns, check our Free Trial FAQ at http://support.na.square-enix.com/faqarticle.php?id=5382&la=1&kid=70929&ret=main