MEL in June is cold (by Ausie standards) and damp, with short days. The Queen's birthday long weekend is generally the official start to the skiing season, but snow coverage is unreliable until July.
AKL in June is similar to MEL, being cold, damp and mostly overcast.
BNE will be warmer with mostly clear skies and cool nights. Still not beach/swimming weather for us locals, but many southerners migrate north to BNE and Gold Coast at this time of the year as it is still swimming/beach weather for them.
Consider buying a domestic fare to the Whitsunday Islands. HMI (Hamilton Island) is available non-stop from SYD or BNE and is far enough north to still enjoy swimming, diving, snorkling etc on the Great Barrier Reef in June.