Unfortunately, there is little chance you'll be able to find a birth record for a ~1804 birth, especially in early America. Some places did them, some didn't. I would reach out to the towns in the area, as well as the churches in those towns. If they have any records there, they will likely do a search for her birth record, however, it is very likely that it does not exist.
Also, how do you know she existed if there's no proof?