I doubt the question will be put to the Australian people again while the present queen is still alive, but the issue may be raised again when Charles becomes king.
Of course, if the question is put again, the issue is not just to dump the king, but whom Australia would replace him with. As I understand it, one of the reasons why republicans lost the 1999 referendum was that many voters were not satisfied with an indirectly elected president as proposed in the republican constitutional amendment. On the other hand, Australian politicians disliked the idea of a president elected by popular vote, as they feared a French-like scenario where a partisan president with a popular mandate of his/her own could clash with a prime minister and cabinet of a different party that held a majority in the House of Representatives. That, of course, would radically change the way Australia has been governed since 1901, violating the "minmalist" approach to bring about a republic with minimal change to the Australian constitution.
As far as I can tell, Australian republicans haven't sorted out that fundamental contradiction yet.