Can you help to solve this mystery?

I see what you mean - it looks as if she died before she was born! However, I think this is probably explained by the fact that deaths had to be registered within five days, whereas six weeks were allowed for registering a birth. The indexes show the quarter when the birth was registered, not the quarter when the child was born. Therefore a child born in March or the latter half of February may be entered in the index for the June quarter of that year.