If you're initially applying for Medicare, and your birthday falls on the 1st of the month, then your Medicare coverage will kick in on the 1st of the previous month. More HERE.
If your birthday is on the first day of the month, your coverage starts the first day of the prior month.
Example
Mr. Kim's 65th birthday is July 1, 2018. If he signs up for Medicare in March, April, or May, his coverage will start on June 1, 2018.