Is there a Gnome Shell shortcut to minimize a window?

Gnome Shell actually supports this. Super+H is the default key. If you want to change the key go to:

Settings -> Keyboard -> Windows -> Hide Window

Since this answer has gained some traction it is good to also know that:

1. This works on MacOS as well.
2. This does not work on Windows, however, you can achieve the same effect by using a tool like AutoHotkey.

I know the OP didn't ask for it but for the sake of completeness, Super+D minimizes all windows (think of it as "show Desktop"). Ubuntu 19.04, Gnome 3.32.1.

On the GNOME desktop environment, you can use CTRL-ALT-D to minimise all and give focus to the desktop. You can also use ALT-F9 to minimise the current window. I think these are the same for Ubuntu/Unity.

Using gsettings:

$ gsettings get org.gnome.desktop.wm.keybindings minimize  
['<Super>h']

Set Super+Shift+H):

$ gsettings set org.gnome.desktop.wm.keybindings minimize "['<Super><Shift>h']"