Do NOT camp at Beenleigh. I can't stress this enough.
Jacobs Well is closer, but how fond of sand fleas are you? You'd have to be seriously fond of sand fleas and love them to the ends of the earth, to even consider camping there ... although it is a nice spot if you are a boatie. Jetty very close by and the smell of salt in the air.
Why not do a search on the net for a scout camp in that general area? I know of 2 and they'll be cheap and cheerful and not that far away.
All other camping options (excluding CVP's) are up to 1/2 hour plus away, but you'd be in a bush setting and possibly near some water.
Thunderbird Park Tamborine MountainTamborine Mountain CVP Scout Camp at Guanaba
Spring Gully Stays - 14k south Canungra
Canungra Show Ground - it's near the pub
Sharp Park / River Bend Country Camping - it's also right opposite another scout camp so that's a third option in exact area - 4k east of Canungra
There are others, but they are further away again.
Kit_e
P.S. How big is the olds back yard?
P.P.S I'd offer up my backyard, but there is no way into it for a vehicle of any kind (shopping trolley excepted).