If you are turning 65 this year, this DOES NOT apply to you. You can apply 3 months, prior to the 1st of the month that you turn 65. Example, if your DOB is 9/15/1957, then the first day you can apply is 6/1/2022. Your coverage will begin on 9/1/2022. You do NOT need to apply yet.
If you have determined that you are going to enroll in Medicare, with coverage to begin on 9/1/2022 (from the prior bullet point), then you should absolutely apply on June 1 and the simple reason is that there are many steps involved, and you are not in full control of some of them (administrative red tape). The list of items can surprise you, and if you wait until August, then we may not have the time necessary to assist in resolving these issues.