In general Microsoft will only support the transfer on an OS from one computer to another when both systems have identical hardware.
As part of the activation process a hardware identifier is created based on the system components, so Microsoft will be able to tell if an OS is being installed on different hardware.
This is not accurate. The computer with Windows 7 on it is a laptop, this being the case I guarantee this is a OEM computer. A OEM copy of Windows is intended for one installation on one computer, period. Once this installation is made it is tied to the computer it was installed on, specifically it is tied to the motherboard.
If you install it on your desktop you will not be able to get it activated because it is tied to the other computer. Microsoft will recognize this and not allow activation.