The most common and most likely answer to this crossword clue is the 5 letter word SHUSH
I've seen this crossword clue in The Times
S
H
U
S
H
Buddy explains!

The straight part
We know that "Quiet" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case quiet = shush.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 SHUSH mean?
SHUSH verb- To be quiet; to keep quiet.
- To ask someone to be quiet, especially by saying shh.