Qt doesn't show
Solution 1:
It is not supposed to be "shown" anywhere.
If you run apt policy qt5-default, you'll see that the package is installed.
The package contains 3 files:
/usr/lib/x86_64-linux-gnu/qtchooser/default.conf
/usr/share/doc/qt5-default/changelog.Debian.gz
/usr/share/doc/qt5-default/copyright
It also depends on qtchooser package that contains some executable files. But there is no .desktop launcher to be "shown".
You can see it at https://packages.ubuntu.com/focal/qt5-default