Re: J. Madison Carver - Born Pickens County
"Most of the Carvers who settled in Pickens Co. were descended from William Carver, Sr. (1740-1819) in Person Co., N.C."
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"Three sons of William Carver are thought to have migrated to Pickens Co., Ala. in the 1830's."
Source: The Heritage of Pickens Co., Ala.
This book contains six articles specifically on the Carvers, plus many more mentions in related family articles.
I hope this gives you a start on where to start researching.