No, just moving the drive is not a good solution, it will occasionally work but most of the time will just blue screen due to incompatible drivers unless everything is very similar.
The Windows 8 license is transferable and if you have a retail disk and key no problem, but if it is OEM build it is not an easy task. You need install media and the retail that can be downloaded won't work. You need the ISO image for 8.1 is the version that has this title: en_windows_8.1_with_update_x64_dvd_4065090. You also need to get your Win 8 key from the bios, which is pretty easy using the command prompt with this command in an elevated prompt (right click run as admin) type it as one long string:
powershell "(Get-WmiObject -query ‘select * from SoftwareLicensingService’).OA3xOriginalProductKey"
The response (once you get it typed correctly) will be your OEM bios embedded key.
If you either updated to 8.1 from 8 or are still using 8 without the update, the image will be different.