And I was just looking, too!
There are three more John Bertrams on the Qld death index. D 1884, father Henry Bertram. D 1926, father William Bertram. D 1889, father Andrew Bertram. Any of these sound like yours, Chris?
I believe Ancestry are about to release records of a large number of free settlers who arrived in NSW. It would be worth checking those, just in case he did arrive in NSW.
Oh, and yes, welcome to Rootschat, Chris!
MarieC