No, no HDD upgrade will give you better FPS.
Upgrading to a Solid State Hard drive would allow games with large maps to load faster, as well as windows and programs being much more responsive.
If you have a 5400rpm or an old ide hard drive then upgrading to a modern hard disk drive will increase general system performance but as stated it will do nothing to help FPS.
GPU, CPU, RAM in pretty much that order is what will help FPS.