Hi Hadi,
The icon on your desktop is a shortcut, i.e. a .desktop file. It should work on most common Linux distributions, but we have not tested it on Mint Linux.
Can you try to open a terminal and lauch QuantumATK from there? You would usually do:
$> /home/USERNAME/QuantumATK/QuantumATK-VERSION/bin/quantumatk
with USERNAME and VERSION matching your setup.
If this works, the program runs fine but something is wrong with the shortcut file. You can create your own shortcut file (see e.g. https://forums.linuxmint.com/viewtopic.php?t=269281), which can also be installed system-wide using the xdg-desktop-icon(https://linux.die.net/man/1/xdg-desktop-icon (https://linux.die.net/man/1/xdg-desktop-icon)) and the xdg-desktop-menu(https://linux.die.net/man/1/xdg-desktop-menu (https://linux.die.net/man/1/xdg-desktop-menu)) tools - these are the ones the installer tries to install shortcuts with.