Too bad I am out of the country at the moment, I would have been happy to show you around. If you are only there a short time, the easiest to access and some of the best riding is at the Smithfield MTB park. very close to Cairns and easy to get around. There are multiple XC trails/loops from easy to lung burning and simple to highly technical.
Atherton is a close second, only because it is furhter out. Maybe 90 minutes from Cairns. But the riding there is just as good, but different. It is more "man made". Nice, flowy trails, longer climbs and loops, more trails, lots to explore.
Then there are epic rides through the depths of the rainforest, further north or south along the coast, each about 60-90 minutes outside of Cairns - Gorrel Track down towards the Cassowary Coast. Or up towards Port Douglas you've got the Bump track and the Twin Bridges track. All of these are simple and basic from a technical MTB perspective, but they are nice long epic rides through some of the most beautiful scenery ever. You feel lost in the rainforest.
Good luck and have fun!