I could give you a lovely list of grown up restaurants following a very decadent childfree w/e DH and I spent in Melbourne in the Summer but unfortunately they won't meet your needs as they were all chosen with the specific reason that there wasn't a child in sight
I was amazed at the Italian influence in Melbourne so you shouldn't have any problems. My friend who lives there with a 9 yo often eats out around St. Kilda. If you manage to escape the sprogs for an evening or lunchtime I would really recommend the original Movida. Mouthwatering. You would probably get away with taking older kids (say 8 plus) there if they like trying different things. My 12 yo is desperate to try steak tartare after my rather drooling description of the Movida version.