If you are referring to Pages '09 or earlier templates, they are stored here:
~/Library/Application Support/iWork/Pages/Templates/My Templates
When you installed Pages v5 or later for the first time, it would have attempted to convert these templates into the User Templates location you mentioned. There are features in Pages '09 and earlier that Apple has never implemented in the newer Pages releases, and that means that some templates that contained those unsupported features were not converted, nor should they be copied now.
I would be careful just copying unqualified templates into that Pages v14.4 User Templates location.