Question

in which Islamic month and day Prophet Mohammad was born and died

Answer 1

The Prophet Muhammad was Born in Islamic Month of Rabbiul Awal The date is 12 according to Islamic Month .The birth date is 12 Rabbiul Awwal.
And He died on the same day.
Hope it helps you!

Answer 2

First you need to understand that the Islamic calender "Hijri" which started with the migration of prophet to "Madina" didn't exist during his birth.S.A.W was died on 8 June 632 AD, Madina, Saudi Arabia.