I place the gmic_gimp_qt file into the gimp plugin directory (either the system wide one, or the user one) but there is no gmic entry in the gimp menu.
That does not look correct.
You download the zip: gmic_3.6.1_gimp3.0_debian13_trixie_amd64.zip
Unzip and it is a folder gmic_gimp_qt containing the plugin gmic-gimp_qt and a Readme. The plugin should already be executable but check anyway.
Put the whole folder in your gimp 3 plugins folder. ~/.config/GIMP/3.0/plug-ins