As it was not a viable option to do it myself, I used Britt's Pet Taxi to pick up my dog from Sydney Quarantine and book and transport him to the airport for a flight to Brisbane. Having checked the price of freight with the transport company myself, I feel Britt's price was very reasonable for her time and service. Britt is a lovely person and a pleasure to deal with, I would recommend her business to anyone!
(Note : While I originally found this service searching Google, I also had other transport companies that weren't able to do a direct pick up from quarantine recommend Britt's Pet Taxi to fill the gap. With Britt's experienced advice we were able to organise things much cheaper and minimise the time my dog spent away from home, after a 30 day stay in quarantine.)

Approximate cost: $480