I’m turning 70 July 1, 2021. Can I request my retirement benefits to begin June 2021, in order to get my maximum benefits?

Hi. Yes. Social Security considers a person to have reached their next age on the day prior to their birthday (https://secure.ssa.gov/apps10/poms.nsf/lnx/0200302400). So, if you were born on July 1 2021, Social Security would count you as being age 70 in June 2021. And, you'd then be eligible for your full age 70 rate if you claim your Social Security retirement benefits effective June 2021.

Best, Jerry