No. Social security does not pay partial months, so you don't actually get a benefit until the end of the first full month you are eligible. Thus, in the scenario we have been discussing, if you turn 62 in early August and then die before the end of the month, you will never start benefits. You also don't get paid social security for the month in which you die. So if you died in September, you won't get benefits. Your survivor is required to report your death to the SSA and return any money received for the month of your death or thereafter.
But let's assume everything goes well. You turn 62 in August. You will not receive any benefit for August.* You will start accruing benefits on September 1 and you will get a check the second Wednesday in October. That check is the payment of your September benefit. Your benefit for October will be paid the second Wednesday of November, and so on. (i.e. - payment in arrears)
* Except if you were born on the 1st or 2nd, then you get credit for a full month.
Here's a site that goes into greater detail --
https://www.simplywise.com/blog/when-do-social-security-benefits-begin/