@martinbaker2727 - Yes, you can. My suggestion would be to follow these steps:
Turn off your target PC. Shut down, power off, the works.
Unplug the least important device with a SATA connection at the motherboard. Start with the DVD drive, if available. If the only thing available are hard drives, pick one that is used for video backup or game storage. You do not want to unplug the drive that contains the operating system.
Plug your hard disk into the freed up slot.
Boot the target computer. It should recognize the presence of a new drive and automatically mount it.
Transfer your data.
Shut down your computer. Power off, the works.
Unplug your hard disk and reconnect the device that you removed in step #2.