Q-ness is an attribute that a Q should have. According to Q, 'Q-ness' is an ideal that is too difficult for a Human to understand.
After Q Junior gave a speech to show what he had learned to Kathryn Janeway and Q in 2378, Q thought that the speech did not show any 'Q-ness'. (VOY: "Q2")
Q describes Q-ness as what he thinks is exemplary in Q standards.