Little bit late tonight, but I promise to you I'm going to explain the details:
Quick steps:
$ WINEARCH=win32 WINEPREFIX=/home/$USER/GarminExpress winetricks d3dcompiler_47 dotnet472 vcrun2010 corefonts
$ WINEARCH=win32 WINEPREFIX=/home/$USER/GarminExpress winetricks win7
$ WINEPREFIX=/home/$USER/GarminExpress wine-stable explorer /desktop=garmin,1366x768 /home/$USER/Downloads/GarminExpress.exe
This works with the latest version Garmin Express 7.1.0.0.
Have a nice day.