Short version:
It's possible to do as you want, but it may not be the best approach in your situation.
Long version:
If you want to convert from a BIOS-mode to an EFI-mode boot of Windows, see this blog post. (There are other pages with similar information out there, too.) Note, however, that this page was written with the assumption that you just want to switch from BIOS-mode to EFI-mode booting without changing the hardware. You want to change the hardware, too, which introduces new complications. Windows really doesn't cope well with such changes, although intermediate to advanced Windows users are pretty skilled at making it work. Personally, I'd be more worried about that aspect of it -- but then again, I'm an expert on EFI-mode booting but not on Windows, so that may reflect my own skills and weaknesses more than anything else.
In any event, the process is basically two-fold:
- Do the MBR-to-GPT conversion. This is fairly straightforward if you're lucky, but could get ugly if you use Windows LDM (aka "dynamic disks") or if your final partition ends without a gap of at least ~33 sectors unallocated at the end of the disk.
- Install an EFI-mode boot loader. This task involves running a number of Windows tools to get the job done.
Both tasks involve some preparation and safety measures, like preparing backups and preparing boot media to use while the system is unbootable.
Instead of going through this hassle, you might want to consider backing up your personal data, doing a fresh install of Windows on your new computer, and then restoring your personal data afterward. This procedure is also a hassle, but it's less likely to go badly wrong, and it will save you from the hassles of getting Windows to cooperate with its new hardware, too.
Another alternative is to use the new computer's Compatibility Support Module (CSM) to boot in BIOS/CSM/legacy mode. You'll still have to deal with the hardware changes, but you won't need to cope with the partition table or boot loader changes.