quiz (one) on (something)
To test one's knowledge about something. Would you mind quizzing me on corporate tax law? I have an exam tomorrow. The teacher stood me up in front of the whole class and quizzed me on the periodic table of elements.
quiz someone on someone or something
to give someone a quiz or test over the subject of someone or something. The teacher quizzed the students on the chapter she had assigned for homework. I hope they quiz me on George Washington. I am prepared.