Your coverage should start on the 1st of the birthday month
See
https://www.medicare.gov/sign-up-cha...coverage-start
where it says "If you sign up for
Medicare Part A (Hospital Insurance) and/or
Medicare Part B (Medical Insurance) during the first 3 months of your Initial Enrollment Period, your coverage starts the first day of the month you turn 65"
Even if birthday is late in the month. Here is their example:
"Mr. Green's 65th birthday is July 20, 2020. If he signs up for Medicare in April, May, or June, his coverage will start on July 1, 2020."
BTW, Social Security does the same. Nothing is pro-rated.
Of course even Medicare is known to make mistakes, but generally they correct them when necessary.