Babar's mother is an unnamed elephant that gives birth to Babar. After she is murdered by the hunter he is adopted by Madame. She is Pom, Alexander, Flora and Isabelle's grandmother and Badou's great-grandmother.
In Babar Returns she is mentioned by Cornelius telling Babar how she had suspected when she was alive and that she'd be proud of him.
Just like in the book, she gives birth to Babar in the old kingdom. In both the series, we don't know who her husband or Babar's Father is. Also, in both the book and show Babar's mother is killed by the hunter and not long after she passed away, Babar was separated from the herd.