This is a very difficult situation. If you do something independently, your daughter is likely to feel that you have over-stepped or that you may be even siding with her daughter against her. If you do nothing, you will feel like you haven’t given your granddaughter the help she needs. You are caught between your loyalties to both. This is never good.
I think the way out of that tricky spot is to talk to your daughter before making any further assumptions or decisions. You’ve always been close so why not try to work as a team to get to the bottom of what is troubling a girl you both love?
If you are concerned that your granddaughter will see it as a betrayal, talk to her beforehand. Tell her that it isn’t healthy for you to be trying to solve a problem behind her mother’s back. She asked for your help and you are going to see that her mother pays attention to the issue — which is the real point after all.
Then see if your daughter will agree to some family therapy — both to help your granddaughter with the cutting and also to help mother and daughter have a better connection. Your granddaughter is only 13. If they can establish better communication now, she and her mother will be better able to navigate the difficult teen years.
I wish you well.