The Liberal National Party on Saturday reiterated a call for a royal commission into the Palaszczuk government's integrity before taking aim at Ms Enoch.
The LNP said Ms Enoch's position was "untenable" after she told parliament last March that former state archivist Mike Summerell had decided not to renew his five year contract and would move to New Zealand.
Mr Summerell this week claimed he had been forced out after his independent statutory role was "compromised" for years by potential government interference.
The state government has since said Mr Summerell was offered a contract extension, albeit only for another three months.
"Leeanne Enoch has clearly misled the parliament. She stated during the estimates hearing and as recently as the last few days emphatically that the former archivist left on his own terms," Shadow Housing Minister Tim Mander said.
"He has now come out and categorically stated that that is not true.
"The only extension that Enoch offered to him was a few extra months to clean out his desk.
"Leeanne Enoch's position is untenable and the premier must sack her immediately."
Mr Summerell has joined calls for an inquiry into the Palaszczuk government following the recent resignations of Integrity Commissioner Dr Nicola Stepanov and Crime and Corruption Commission chairperson Alan MacSporran.
Dr Stepanov, who finishes her role in June, and Mr Summerell have both complained of "interference" in their roles.
Asked about Ms Enoch on Saturday, Treasurer Cameron Dick said: "I am not aware of any of those matters and that is a matter for Minister Enoch".
LNP leader David Crisafulli said Mr Dick's "failure to back" the Housing Minister was "damning".
"There are serious questions about the way ministers are conducting themselves," he said.
Mr Crisafulli wrote to the premier this week asking her to call for a royal commission.
But Mr Dick said Queensland had a "very robust" system.
He cited a Parliamentary Crime and Corruption Committee report released in December that called for a royal commission into the Queensland CCC that is being considered by cabinet.
Mr Dick believed Queenslanders also needed to be reminded that a five-year strategic review of the Integrity Commission by Kevin Yearbury has been referred to a parliamentary committee for consideration.
"I don't know whether Queenslanders are fully aware of that. I haven't seen too much reporting on that," he said.
But Mr Crisafulli said it was "not dealing with a review that will go into the heart of systemic corruption in the Queensland government".
"This is a call for accountability," he said.
However Mr Dick said "being lectured to by the LNP on integrity in government is like being lectured to by a drunk on sobriety".
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