Public buses in Cairns seem to be operated by Sunbus, but they're part of the wider Queenland Translink system. There's a special Translink website area for Cairns, and a lot of the links from the Sunbus site for information take you back to the Translink site. The outside of the buses I've seen in Cairns all have Translink written on them.
Within the South-East Queensland area, it's generally recommended to get a Go Card to pay for your bus / train / etc trips. (It's much simpler and cheaper than using paper tickets). The Go Card site says
go card is TransLink's South East Queensland electronic ticket. It allows you to travel seamlessly on all TransLink bus, train, ferry and tram services.
Cairns isn't in South-East Queenland, but it does have buses run on behalf of Translink (with Translink branding on the side), and the Translink page on ticket pricing covers both Cairns and SE-Queensland.
So, given all that, is a Go Card valid on buses in the Cairns area?