She may have a brother, since James T. Kirk said he was staying at his uncle's farm in Star Trek Generations. It was never made clear which of his parents had a sibling though, or if he was just using the term for a long time family friend. In Star Trek, an uncle Frank was cut from the script and reworked into Winona's new husband heard in the film.
Information from several of the novels and short-stories states she was born in 2210 and was of Sioux descent.
In the TOS novel The Janus Gate: Past Prologue, Winona accompanied George to assignment on planet Grex is 2247. They left the planet when a civil war broke out one year later and moved back to their farm in Riverside.
In the novels Enterprise: The First Adventure and Final Frontier, Winona left Earth in 2259 after the death of her husband to stay with her eldest son in Deneva. She was present during the change of command ceremony when Christopher Pike passed command of the USS Enterprise to her son James T. Kirk in 2264.
According to Time for Yesterday, she died in a fire on the family farm in Riverside, Iowa on June 2, 2283. She was cremated and her ashes were placed in a crypt in a chapel at her hometown of Riverside.