Hi Zack,
The quick answer is "no, this is not possible" (at the moment).
The long answer is that - for me - the kind of questions that you would ask in a quiz are totally different in their nature from the questions you ask in a survey-type activity such as Questionnaire (or Feedback). The former (quiz) will get from the respondents answers which will be judged as "correct" or "incorrect", whereas there is no notion of correctness in a survey. Therefore I do not see how that feature might be useful.
However, I agree it would be useful if the Questionnaire (and the Feedback) modules offered a "questions bank" feature similar to the one in the Quiz module. This has been asked in CONTRIB-1255 and anyone is welcome to vote for it.
Joseph