The most likely contender for this discarded egg is the Crested Pigeon (Ocyphaps lophotes). This species has been seen nesting in these trees, and the only other species seen nesting here are Noisy Miner (Manorina melanocephala), but their eggs are speckled. Egg size approx. 3cm end to end. Image searches for this species and its eggs lead me to think this spotting ID is correct.
Found in the grass under a tree, but I couldn't find a bird's nest above.
A friend spotted this egg, and knowing that I'm a nature nerd she said I should have a look. After taking a few photos I showed her and a few others as well. The shell was cracked but I thought I could also see what appeared to be a beak and possibly some feathers as well. Some said it was a lizard, others a snake - there was something on the shell but it was hard to ascertain exactly what. As it turns out, it was just some grass glued to the surface (difficult to tell on a small camera screen). I held the egg up towards the light, but there was no shadow within and therefore no embryo. I cracked the brittle shell open and there was only the yolk. Most certainly a bird egg.