I think you can use a site called web jet to compare flight prices, or you can go onto the websites of individual airlines. If I'm travelling within Australia, I tend to use Jetstar or Virgin. In their sites, you can select the dates you want to travel on, and then it gives you a selection of times during the day to do it, with the prices. Be careful though, if she's got luggage to bring, that you don't select a fare that only allows carry on luggage (which is usually just one small bag that you take on the plane with you.)

I think it takes about two hours from Brisbane to Melbourne, if my memory serves me correctly. And you don't need a passport, that's only for overseas. Hope that helps