She had four grandchildren, three from her son George Samuel, and one from her son James, as well as one miscarried child. (TOS: "What Are Little Girls Made Of?", "The Paradise Syndrome"; Star Trek II: The Wrath of Khan)
Background information Edit
In the script for the ultimately never-produced film Star Trek: The First Adventure (dated 1 August 1989), Kirk's mother was named "Joan". The name "Winona" originated in Vonda N. McIntyre's 1986 TOS novel Enterprise: The First Adventure, and became canon with its inclusion in the 2009 film.
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.
She may also have been the grandmother that Peter Kirk could have lived with, had he so chosen, after his events on Deneva. This individual was mentioned in a deleted scene from TOS: "Operation -- Annihilate!". 
In the TOS novel The Janus Gate: Past Prologue, Winona accompanied George on his assignment to the planet Grex in 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.