Due to macOS Catalina lacking support for 32-bit applications, you may wish to continue using an older versions of macOS or just have a dual-boot install with an older version. Either way, having it nag you to upgrade to Catalina (when you may already have it installed on your system) can get annoying.
Use the following method to disable the Catalina upgrade notifications.
First, open Terminal, then run the following command:
sudo softwareupdate --ignore "macOS Catalina"
To remove the notification dot from System Settings, run this:
defaults write com.apple.systempreferences AttentionPrefBundleIDs 0 && killall Dock
To un-do the above change, run this in a Terminal window:
sudo softwareupdate --reset-ignored