No, since the patient is a child, it's not a HIPAA violation to tell her mother that you were her nurse. If she was 18, it would be different. However, like others said, why bother bringing it up? Perhaps her daughter's hospitalization is not something she cares to discuss after the fact.
In my last job, I was a visiting nurse primarily in my city. I had patients who lived less than a mile from me. Occasionally, I'd run into them in public. If they said hi to me, I'd say hi back but I never initiated any conversations with them in public.
When I worked in the hospital, I took care of my 8th grade gym teacher's daughter. I recognized her right away but it wasn't until she said "do I know you from somewhere?" that I told her where she knew me from.
I also had the opposite experience. When I was in college, I had a rather severe eating disorder and major depression. When my primary nurse practitioner went on maternity leave, a second nurse practitioner joined the practice. Because of the nature of what I was seen at that practice for, she knew things about me that I have never discussed with anyone in my adult life. The last time I saw her as a patient, she sent me to the ER on a suicide hold. Fast forward several years later and this nurse practitioner took a position on the floor I was working as a staff nurse on. She recognized me IMMEDIATELY but couldn't place me. She asked me if I had been one of her students. A few days later, someone mentioned her last name and I then immediately knew who she was and why she recognized me. It bothered me a lot that someone at work now knew things about me that I never cared to discuss again. Neither one of us mentioned anything else for several months, until something came up about my home town and when I said "that's where I'm from", she said, "I know." Then I asked her "oh, so you finally remembered where you knew me from?" Then she told me that she'd remembered as soon as she asked me if I had studied with her but, since she realized I had been her patient, she was horrified that she'd even asked and had no intentions of ever bringing it up again.
I think that's the best approach to take in this case. Don't bring it up but if your friend does, you can say "yes, that was me."