You could configure sudo without password only for the wg-quick command. Not a beautiful solution and also not super easy to configure with sudo but it’s an option. Much easier solution is to install doas or opendoas and leave sudo as it is. Let’s assume your desktop user is named variants and in group variants, then putting the following in /etc/doas.conf is expected to work :
permit nopass :variants cmd wg-quick
If your user is not yet in a group, and you add a new group, remember that making such a change usually requires logging out and logging in again because the group changes become visible. Then test with doas wg-quick up wg0