When it comes to birth records in Monroe Co., MI you have two places to locate them depending on the time period that you are looking for the person. They are the Monroe County courthouse in Monroe and the state library in Lansing.
Birth records at the courthouse run from 1874 to present. This is where you would be able to get certified copies for this time period. As with all places of this type be patient with the staff and allow for some time with your search.
The Michigan vital records start for births in 1867. I do not believe that this applies to Monroe Co. births since state records normally reflect what is on hand at the county level.
Birth records give you excellent first hand information on time and place of birth. This is also the best source for parentage. Always make copies of these records for proof and verification.