Hi @AlenaStehlickov ...
There really isn't a way to add Turnitin to Quizzes (Classic or New) at this time. I did find one topic here in the Community that is somewhat related, and so I wanted to point you to:
Solved: Turnitin and Quizzes - Instructure Community (canvaslms.com)
There is also this existing Feature Idea:
New Quizzes: Allow use of turnitin - Instructure Community (canvaslms.com)
I'm not sure if either of these will be helpful to you, but I wanted to pass along the links just in case.
I know this wasn't probably the answer you were looking for, but I hope it helps to answer your question.
