While comprehensive reporting is yes lacking, Weimar in Vic was reported by the ABC yesterday as saying:
"Less than half the people, 47 per cent of the people in the hospital were not fully vaccinated," he said.
"Three-quarters of those in ICU, 74 per cent, were not fully vaccinated."
and notably
Mr Weimar added that there is "nobody in ICU who has had a third dose".
So one would assume that they are tracking vaccination status (ie no, single or triple dosed, even if they do not make public all the data.
Mind you as the % of people who have had three doses is only just over 20% now that data at least in Vic would be quite limited.