I own both an Alienware laptop and a custom built PC, I have no problem running GOG's version of Fallout 3 on my PC, but every time I click "New Game" on my Alienware, the game crashes and it's really been frustrating me because I thought after dealing with the bull shit of the Steam version I wouldn't have a problem, are there any suggestions to fix the problem?

I don't think there's any problem with it being on Windows 10 because my desktop has Windows 10 aswell. One of my friends told me it could be my Intel graphics card creating a problem but I'm not sure.

Check this.

This didn't work for me are there any other solutions to this same problem?

Have you tried this fix? https://www.nexusmods.com/fallout3/mods/21888/?
The GoG setup vid for it is here
https://www.youtube.com/watch?v=QkIKRsArlSU&t

Also try running as admin.