The Melbourne International Exhibition (the 8th World's Fair) was held from 1 October 1880 until 30 April 1881. I couldn't find a report of a shotgun shooting competition but firearms were on display.
The first "Queen’s Trophy" rifle competition was held at the "Victoria Ranges" in Williamstown.
The 'Official Record' is here
https://archive.org/details/bub_gb_0csNAAAAQAAJ