(Not a teacher)
The only way I can think of 'go blue' as an idiom is as either to show someone is cold, as you said, or meaning that someone can't breathe (during which they likely do turn blue-ish, and so it isn't really an idiom).
The only 'colour' idiom I can think of that relates to examinations is 'pass with flying colours' meaning 'pass with a high mark'.
But what the instructor said, or what your friend heard, I have no idea what it means in relation to examinations.
Student or Learner