Sturgeon was First Minister when the hospital opened and Health Secretary when it was commissioned.
This week NHS Greater Glasgow and Clyde said it was under pressure to open on time and on budget and said it was probably opened too soon.
Labor and the Conservatives this week put pressure on John Swinney to explain who put pressure on the health board.
Today the mum of 10year-old Milly Main, who died at the hospital after contracting an infection linked to the water supply, urged Sturgeon to be honest and come forward about what she knew.
Kimberly Darroch said: “I do believe Nicola Sturgeon knows something. My message to her is to come forward and be honest.”
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The former first minister has now said it is untrue she was aware of any safety concerns over the hospital.
Nicola Sturgeon said: “I understand Ms Darroch’s determination to get answers, and my thoughts are very much with her and others affected by the situation at QUEH. I hope the Inquiry report, when published, will offer those affected the answers they want and deserve.
“Any suggestion that I applied pressure for the hospital to open before it was ready or that I had any knowledge of safety concerns at that time are completely untrue.
“The families who lost loved ones in the QUEH have a right to have their questions answered. That is why the independent public inquiry was established while I was First Minister.
“The Inquiry has had sight of all relevant material and has also been able to take evidence from any individual with insight or information it thought necessary for its work.”
