What should I do with the X11 Security folder after upgrade to Catalina?
I am surprised that no one has answered it. 

If XQuartz was installed prior to installing Catalina then this X11 folder will exist. Uninstall your version of XQuartz using this gist (ideally before upgrading macOS). However, you have already upgraded to Catalina so the X11 shortcut folder can't be deleted as it appears in your desktop as a relocated item. If you want to know more, you can check the pdf folder too. 


  You can find the Relocated Items folder in the Shared folder within
  the User folder. The PDF document in the Relocated Items folder
  explains what to do with files placed in this folder.


Here is a solution that you can do on your side:


Reboot and go to Cmd+R recovery mode
disable SIP by csrutil disable
Reboot to normal mode and delete that folder, empty trash
Reboot and go to Cmd+R recovery mode
re-enable SIP by csrutil enable
Reboot to normal mode