Generally No, you cannot use expenses incurred before your service date as eligible expenses.
That being said, I would recommend calling your FSA administrator to confirm, as there have been a lot of leniencies granted recently for Dependent Care FSAs due to the pandemic.
A Perfect Example: last year I was in a similar situation as you because my children were in daycare at the beginning of the year, but we didn't have a DC FSA until my wife took a new job in Feb, 2020. The kids left daycare in March and we would have had over the $5K max from the beginning of the year, but not if we started from my wife's first day of eligibility in Feb. HR confirmed we couldn't use receipts from before then so we filed the change of status and stopped contributing.