The water supply lines are 1" in front of the wall where I want to install a new shower pan. Can I just add wood to the face of the wall studs so that the new pan will not hit the supply line?
If you plan to fur out the whole shower wall, then yes, you can. If you only want to add a board to bump the pan away from the wall, then no, you can not, as you will be creating a "designed leak". You always want water to shed directly into the pan, you don't want to be building shelves, troughs, or passageways for the water to sit or meander through before it gets to the pan to drain away.
The caveat to this is if you are installing a hot-mop pan, then you could conceivably just bump the pan, but then you'd also have a curb you'll be tripping on at the shower wall. Not a good plan. Furring out the whole shower wall will look and function better, regardless of the type of pan you use.
Spruce has the proper suggestion for your situation.
Another option is to by pass this 1" pipe somehow, so you don't have a need for such an involved solution. I can't see what you have, but relocation of the pipe could be possible.
Whatever you decide to do, do it in a manner to avoid future complications and leaks.
sounds like maybe you took out and older one piece shower unit? there was probably a floor joist under wall. I have seen and done this to avoid drilling joist.