The Self-Government Act governs no-confidence motions. Kate Carnell is both Treasurer and Chief Minister.
A no-confidence motion passed by the Assembly against a mere Minister (such as Treasurer) of itself has no legal force. The Self-Government Act says only that a Minister may be dismissed at any time by the Chief Minister.
The only force a no-confidence motion against a Minister has is an implied threat by the Assembly to vote no confidence in the Chief Minister unless she dismissed the Minister. So when no-confidence was voted in Sports Minister Wayne Berry in 1994 (check), he had to resign or be dismissed otherwise then Chief Minister Rosemary Follett might have faced a no-confidence motion.
That carries more weight, because the Self-Government Act says if a motion of no-confidence in the Chief Minister is passed, all the Ministers vacate their offices.
If Carnell were Treasurer only, a no-confidence in her would result in her losing ministerial office, but the rest of the Government staying in place.
But if there is a no-confidence motion in Carnell, it will of necessity be in her role as Chief Minister. If it is successful, all the Ministers vacate their offices.
The success of a no-confidence motion will depend on Paul Osborne and the cross-bench and is not a strong possibility. If passed, the Self-Government Act requires that the members present elect a new Chief Minister. The Speaker stays put, unless voted against separately.
Trevor Kaine (United Canberra) whose disfavours Carnell rather than the party she leads could easily vote against her then vote for the new Liberal nominee for Chief Minister, rather than for Labor’s Jon Stanhope, and, if it came to it, that would be enough to tip the balance.