It will not "damage" it if it is a compatible connection (SATA -> SATA and PATA/IDE -> PATA/IDE). It won't physically interface with the computer if it's an opposing connection type.
Another thing to note is that if it is the correct connection, you may not be able to boot into Windows (even with the necessary BIOS adjustments). The Windows installation from the Toshiba drive will have different drivers than the HP laptop, and so you may be presented with blue screens.