June Carter Cash is one of the Carter Sisters, who were on stage with their mother, Mother Maybelle Carter, the famous woman who played in the Original Carter Family with the brother of her husband....
It would appear not. There's no mention of a relation between the two. GWC was a prominent Southerner who settled in Louisiana after the Civil War. Jimmy Carter is a Georgian.
She was married 3 times and had 1 child by each of her husbands, Carlene with husband Carl Smith, Rosie with husband Edwin Nix and John Carter Cash with Johnny Cash.