Hopefully you have seen the notice in the Timaru Herald (13/8/1878) announcing their marriage on 6 Aug at St. Mary's Church. William Gunn was an ironmonger in Dunedin and Elizabeth was the eldest daughter of the late John Melton of Timaru.
William Gunn went on to become a Dentist. He had a brother in Dunedin, Alexander, and the following link gives the details of their parents in Scotland.
A short biography of William Gunn is at this link (Evening Post, 19/12/1914):
On 23/5/1879 William and Elizabeth Gunn had a daughter, Elizabeth Catherine, d.1963. Her biography is at:
In 1881 they had a son William Alexander
On the 25th Feb 1883 they had a daughter, Ellen Louisa, born and died in Timaru d. 27/8/1943.
In 1885 they had a son, Gordon Melton
in 1888 they had a daughter, Alice Johanna who died in Timaru 10/5/1980.
Elizabeth Jane Gunn d. 20/10/1937, buried 22/10/1937 in Timaru, aged 83.
William Gunn d. 16/12/1914 in Timaru, aged 64. (4 photos available).
There are several references to the 2 boys, William and Gordon, in school sports in the Timaru Herald 22/10/1898.
If any of the links don't work for you, let me know.
Good luck, Kevin