Yes, outdated/incorrect video/graphics drivers have caused more than a few BSOD's. That is usually the first thing you should do when you get a new computer or change operating systems: update all your drivers. You have to go to your manufacturer's website and download the latest drivers for your particular model and operating system. Things should go smoothly after that.
It's good that HP is willing to send you a new computer if you keep having problems, but you should be able to fix things yourself in this case.