The most common and most likely answer to this crossword clue is the 6 letter word SOFTEN
S
O
F
T
E
N
More clues leading to the result SOFTEN
- Cushion frequently seen under bottom of missus
- Cushion, with pins removed, spins over and over
- Become gentle, beginning to soothe frequently
- At the end, drugs frequently mollify
- Temper that kicks off somewhat frequently
- Subdue back of class at frequent intervals
- Moderate son getting a great deal
- Temper beginning to show again and again
- Appease old Bob frequently
- Yield to success usually
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 = soften.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 SOFTEN mean?
SOFTEN verb- To make something soft or softer.
- To undermine the morale of someone (often soften up).
- To make less harsh
- To become soft or softer