The most common and most likely answer to this crossword clue is the 9 letter word WHISPERER
I've seen this crossword clue in The Times
W
H
I
S
P
E
R
E
R
Buddy explains!

The straight part
We know that "Quiet speaker" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case quiet speaker = whisperer.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.
Crossword Answer definitions
WHISPERER noun- Someone who whispers.
- Someone who tells secrets; a gossip.
- Someone who is skilled in taming or training a certain kind of animal, using gentle vocal commands and body language as opposed to physical contact. See horse whisperer.
- (by extension) Someone who has an uncanny ability to control or manipulate a certain thing or person; an expert or guru in a particular field or subject