If this bank statement is issued every three months, then "Please provide a copy of the most recent months' bank statement" is correct, and in effect, means the most recent statement received, covering those three months..
If, as is more usual, it is issued every month, then it should read:
"Please provide a copy of the most recent month's bank statement."
and refers to the most recent statement you received from the bank, covering a period such as 19/7/2015 to 18/8/2015.