That's a tough one. Even in the larger Oz cities there aren't that many services so Cairns is probably woefully under serviced in that regard.
I found this for you but it appears to be five years old...could offer up some clues...
Cairns
SSAY IT (same sex attracted youth, intersex and transgender): A group for LGBTI young people that runs on Wednesdays from 3:30 to 5:30. Contact David at YETI 4051 4927
David@yeti.net.au or John at Youth Link for more information
QAHC LGBT Resource Centre
Good luck!