The latest version of Quicktime should run in Catalina. Apple even has a user guide for it on their support pages. What version is giving your problems? Have a look for QuickTime in your Applications folder to verify that the latest version installed by the Catalina installer does or does not work for you.
I understand that the older version, Quicktime 7, will not run (I don't have Catalina on my Mac and can't test this).
If either version of Quicktime fails on your system there are other options for playing media files, such as Iina Player.