Michigan did not start keeping vital records until 1867 so there are no birth records before 1867 nor death records. There are some marriages recorded before that in court ledgers at the county level.
So to find a birth you need to locate the church they attended. However it depends on what religion because some did not do infant/child baptisms. Once you determine a church then it depends on if those records are still in existence and locating where the records are kept.
So I suggest try to figure out their religion and locate the church.
Another option is to see if it was reported in the newspaper (so and so had a baby...) but I don't think there are newspapers for St. Clair County that early. I can't remember what of the Pt Huron paper exists but it did not cover everything in the county.