The most common and most likely answer to this crossword clue is the 4 letter word EXAM
I've seen this crossword clue in Financial Times
E
X
A
M
More clues leading to the result EXAM
- Without a mark where they should be gained
- Former partner with morning paper
- Test showing reversed cut when Mike’s opening
- Mike cut back paper
- Test in English before ten in the morning
- Once given a first in Mathematics — must have sat this
- Test: most ending in failure rejected
- Set up monitor primarily to eliminate paper
- King mislaid crown before midday inspection
- No longer associated with American test
Buddy explains!

The straight part
We know that "quiz" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case quiz = exam.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does EXAM mean?
EXAM noun- The act of examining.
- Particularly, an inspection by a medical professional to establish the extent and nature of any sickness or injury.
- A formal test involving answering written or oral questions under a time constraint and usually without access to textbooks.
- Interrogation.
- To observe or inspect carefully or critically
- To check the health or condition of something or someone
- To determine the aptitude, skills or qualifications of someone by subjecting them to an examination
- To interrogate