Britain, Canada, Australia, New Zealand, Papua New Guinea, and a dozen or so other countries (mostly small islands in the Pacific and Caribbean) are known as
Commonwealth realms. They're all independent self governing monarchies that just happen to share the same monarch. Each of these countries are free to set their own laws as to what designs appear on their own coinage.
Australian laws, for instance, say that the Treasury department (and ultimately, the Treasurer and other politicians) can decide to put whatever they like on the coins. We keep the Queen there out of respect for tradition. She's still our Queen, and a majority of the population want it to stay that way. Past suggestions to remove the links to Britain still seen on the coinage designs, flag, and other national symbols have proved unpopular, with monarchists accusing such moves as being "republicanism by stealth".
Papua New Guinea, on the other hand, is constitutionally very similar to Australia - the Queen is Queen there, too. However, PNG coins never show the Queen, unless the coin is actually commemorating something royal eg. a royal visit. The obverse of all PNG coins is the same: the country's national symbol, a bird-of-paradise sitting on a drum.
Britain also still owns several "colonies", the last remnants of the Empire of yesteryear. They're called
overseas territories now, though they're still not independent. Most of these countries also make their own coins and their coins always have the Queen on them, since their laws are still largely based on British laws.
Fiji is anomalous in that it became a republic after a military coup in 1987, but they continue to use the Queen's portrait on their coins (though the portrait they use is the now-obsolete one that was current back when Fiji became a republic). I don't know why; you'll have to ask a Fijian.
