If there is only one correct answer, I could use radio buttons. If there are multiple correct answers, checkboxes. However, since it's a quiz and not an information gathering form or survey, I could use checkboxes all the time to make it more difficult. Though this is could be a bit deceiving to the user, what say you?
I think it is only fair to indicate wether you allow multiple answers. You could either state this at the top ("Questions may have multiple correct answers, check all that apply" and then use checkboxes throughout) or state it per question. I would however mention it somewhere, because otherwise you are not making it more difficult, but mostly more random (because interpretations of the UI are more decisive than knowledge, and that makes your quiz less useful. So I wouldn't do that.)
As Inca states, radio buttons and checkboxes attend to different type of answers. Radio buttons will allow for only one choice such as
Choose your favorite food in contrast to the checkbox approach which would most likely serve for
Choose your favorite foods. Note how the second question implies multiple answers.
You are quizzing their knowledge, not their interpretation of a web form. Use radios.