There are few gaming conventions with a primary boardgame focus in Australia. The ones listed below are boardgaming centred and are family friendly. The intent at all of these conventions is to have families and new friends play together, but not to act as babysitters, so check any age restrictions before planning on your child going to the convention on their own. Most video game / role-playing / science-fiction conventions will feature some board gaming, but might not be suited to hopeful game players walking in off the street.
The Australian Toy and Games Expo (Sydney)
This convention is the biggest aimed at casual gamers. It runs over the Queen's Birthday long weekend (mid-June). It is based at Showgrounds (ex-Olympic venue) at Homebush.
There are many retailers as well as small publishers demonstrating their games. Lots of games are available to borrow.
In the evenings, the gaming moves to another venue.
Border Con (Albury-Wodonga)
This convention runs in Albury, on the NSW/Victoria border. It takes place on the Queen's Birthday long weekend (mid-June). It has been run in the Hume Inn Motel, which provides the gaming venue and also offers accommodation.
Con Vic (Melbourne)
This long-running convention generally occurs twice a year in Melbourne, most recently in Laverton. Organized by Julian Clark who runs Unhalfbricking Games, the convention offers a relaxed and family-friendly style. There are usually some informal tournaments as well as lots of open play.
There is a list of some other gaming-related Conventions at this link