The symptoms suggest the 328p isn't running, and your tests confirm it's getting reasonable power, therefore I believe it is dead. You could buy a 328p with bootloader installed on it, or you could try this:
www.electronicshub.org/burn-bootloader-on-atmega328/
I'd never heard of the GQ-4X beforel now. According to Google it will program the 328p if you have something called the AVRISP Connection Adapter. Perhaps you know what that means? I'm looking at
www.mcumall.com/comersus/store/mcumall_T...WillemsupportICs.asp
and searching within the page for Atmega328*.
If you buy one with bootloader preinstalled I don;t think you'll have to mess with the "fuse bits". If you burn the bootloader yourself the fuse bits will need to be configured and I'm not real sure how to do that when using an Arduino as your programmer (I have an Atmel AVRISP Mk.II that I use for this kind of thing, never tried it with an Arduino). In case you need it here are the fuse bit settings:
lfuse = 0xFF
hfuse = 0xDE
efuse = 0x05
As for the code, the clock code and clock-specific libraries are here:
bitbucket.org/isparkes/nixiefirmwarev1
assuming your clock has RGB LEDs, not NeoPixel LEDs. Read the page for more info about clock versions & code.
You will need the Arduino IDE from arduino.cc in order to compile and burn the program. You will also need to add a few general purpose libraries to the IDE. Look at the #includes at the beginning of the clock program to see which libs you need to get.
If you don't feel like messing with all this I'd suggest you contact Ian through the nixieclock.biz site. He can probably hook you up with a ready-to-run 328p for a decent price.