Visit Brisbane during a layover

Brisbane Airport is not far from the city, and the Airtrain service takes you to the city centre in 20 minutes, running every 15 minutes. It's not cheap though: $30 for a return ticket.

Allowing for 30 minutes to get out of the Airport and a reasonably safe 2 hour margin to be back before departure, that leaves you about 3 hours in the city.

Brisbane doesn't really have any big, impressive sights, except perhaps the Queensland Museum. If the weather is good, I'd just cruise around on the river using the public ferries, take a stroll through the botanical gardens and the south bank parklands, and look for a nice place to eat.

As Michael says, the Airtrain will quickly whisk you into the city centre. Book online to save a bit, if you're sure your inbound flight is on time! Try to check your bags through, so you don't need to take them with you / don't need to check them in later, and get your onward boarding pass if you can before leaving the airport. Also, check with the airline when you need to be back to clear security, then add a bit of padding, to work out when you need to head back.

The AirTrain will drop you at Brisbane's Roma Street station. That's right next to a large park (Roma Street Parklands) which I quite like - there's lots of variety there, and some really cool Dragon Lizards to spot.

If there's also time, head over the river to the Cultural Centre on the South Bank. There are several great free museums here, where you can learn more about Queensland, see classical and modern art from the region etc. The South Bank also has nice parks, and if you wanted to stay outside (bring suncream!) you could go down through the South Bank parks, over the Goodwill Bridge, and back through the City Botanic Gardens (which even has free wifi!)