Hi Anne,
There are some Mulqueeneys in the Assisted Immigrants index (
http://www.records.nsw.gov.au/archives/search_1656.asp):
MULQUEENEY Bridget 30 and husband Joshua 1858
MULQUEENEY Dennis 7 and relations Aloe 1857
MULQUEENEY Elizabeth 15 Mary Pleasants 1858
MULQUEENEY John 23 Admiral Lyons 1858
MULQUEENEY John 24 Mary Pleasants 1858
MULQUEENEY Mary 24 Emily 1850
MULQUEENEY Mary 11 and relations Aloe 1857
MULQUEENEY Owen 32 and wife Joshua 1858
MULQUEENEY Patrick 9 and relations Aloe
...also some MULQUEENs and MULQUEENYs.
Could be related?
Prue