Hi Jen,
Welcome to the Forum. I live in Canberra so not all that far from you. Detecting on the side if the road generally puts you within the road reserve and digging holes there is not allowed. Beyond that if you are not jumping fences or on private property you should be OK as much of that land around the Flat is crown land. The Flat itself was mainly a copper lead and zinc mine with only very microscopic gold amongst it.
The Queanbeyan River up near the Flat holds some gold but not generally large enough to detect. Your best shot would be up around the Badja gold field or Oallen Forde in from Braidwood. If you know anybody in the Araluen area then that's a good area as well. Good luck....Wal.