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 2009 film Star Trek, Winona Kirk, specifically the alternate reality version of the character, was portrayed by Jennifer Morrison. This footage of Winona Kirk was set immediately after the split in the timeline, allowing an insight into what the prime-universe version of the character was like, as Winona Kirk has never appeared in canon aside from that.
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" became canon with its inclusion in the aforementioned 2009 film.
Star Trek writers Alex Kurtzman and Roberto Orci revealed Winona, at least as they thought of her, was a Starfleet officer, hence her presence aboard the Kelvin.  She was described as a Starfleet officer in the Star Trek Encyclopedia (4th ed., vol. 1, p. 424).
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!". 
Information from several of the novels and short-stories states that this character was born in 2210 and was of Sioux descent. The 1982 reference book Star Trek II: Biographies gave her name as "Marjorie Wimpole". The name "Winona" originated in Vonda N. McIntyre's 1986 TOS novel Enterprise: The First Adventure.
In the TOS novel The Janus Gate: Past Prologue, Winona accompanied George when he was assigned 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 the novel 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.