I'm going to tell my friend's mom that the autographed picture she gave me is a fake
How did you find out the picture was a fake?
The quality is bad and it doesn't have a copyright mark on it.
Honesty is always the best policy, but volunteering information that will upset your friend's mom or hurt her feelings is better left unsaid. Things that are better left unsaid should remain as thoughts in a person's head because they will cause unpleasantness if said aloud.
I suppose I should keep this to myself.
She probably knew what you wanted but was unable to pay the full price for it.  The gift was at least thoughtful and you should simply thank her without letting her know it was fake.