If the laptop that you purchased came installed with an OEM license key, then no, you cannot use that license on another computer.
OEM license keys are tied to the computer that they are installed on, and cannot be reused legally (unless you are reinstalling on the same computer). OEM keys are on a label sticker on the chassis of the system they are tied to.
However, if the license you have is a retail copy of Windows, you may use that license on your desktop. Retail license keys can be moved, but the previous Windows installation must be removed before installing the new one.
Retail license keys are bought in a box with a Windows installation disc separate from a computer. If your computer came installed with Windows from a store, then this is not what you have. It is an OEM key.
Be aware that none of the advice on Superuser should be taken as 100% accurate legal advice. Always read the EULA for your particular version of Windows for the country in which you reside for detailed information on what you are permitted to do with the license.