I run Ubuntu on my machine, however, I'm unsure if what I did will work the exact same for your OS; if it is Ubuntu-based, it probably should.
To add Godot to the "Applications" menu, where you can search for an application to run, similar to Windows' own Start Menu, you will need to create a *.desktop file and place it in the directory ~/.local/share/applications
. The file will need the following lines at a minimum:
[Desktop Entry]
Name=Godot
Exec=<Path to executable file>
I put a few additional lines, such as Icon=<Path to Godot's icon.png>
, and Type=Application
. I have also set the file to be flagged as an executable file. This let's me find Godot whenever I search "Godot" in my Applications menu, like in the image in this comment:

I'm unsure if this is exactly what you meant, but I figured it would help if it was.