An 1884 affidavit made out by Sarah Brown indicated that she and Thomas Brown were married in 1830 by Reverend Joseph Whitehead in Mercer County, Kentucky.
A search on Google Books for "joseph whitehead kentucky" (sans quotes) located several references, including one to a published set of records from the Bethel Baptist Church in Washington County, Kentucky. It's a good thing I didn't include Mercer as one of my search terms or this item may not have been returned as a hit.
Washington County borders Mercer County. At this point, it is not know where within Mercer County the Brown family lived, so it is very possible they are mentioned in these records or at the very least there may some mention of Reverend Whitehead in the church records that have been published. One has to be careful not to assume residential details that are not known.
Given that these records were published recently, there's no full text available for them on Google Books. A search of the Allen County Public Library's card catalog indicates that the item is in their collection, so it will be added to my list of materials to locate when I'm there in August.
Baptist church records are typically not overly detailed in terms of vital records information, but one never knows what may be included.