Recently i presented SharePoint 2010 permission and visited MSDN Forum people asking about how to convert web application that created with classic mode authentication to claim based authentication.
Yes, you can convert only via PowerShell Script.
$ConvertApp = get-spwebapplication “http://<web application name>”
$ConvertApp.useclaimsauthentication = “True”
Please Note : Once you convert the web application to use claim based authentication, you cannot return to classic mode authentication.