I'm not familiar with GMMK boards, usually you'd use dfu-util to flash the QMK firmware, maybe if you can save/extract the firmware bin file from the exe (dry run it in Wine perhaps?), you could flash it using dfu. But I'd first check if your board supports that.
But you could always just install Windows to a USB stick and boot from it, you don't need to find a separate Windows machine. I mean, you don't even need to even buy a license or anything so you've got nothing to lose here.
And in case you've got an existing Ventoy USB, then you can just install Windows in a VM, and drop the VHD into your Ventoy USB so it doesn't pollute your drive with a bunch of files. You'll end up with a nice and self-contained copy of Windows, just like any other Linux ISO, and you can use it in situations like this without feeling guilty of dirtying your PC.