The most common and most likely answer to this crossword clue is the 4 plus 4 plus 6 letter word HOLD ONES TONGUE
I've seen this crossword clue in The Times
H
O
L
D
O
N
E
S
T
O
N
G
U
E
Buddy explains!
The straight part
We know that "to keep quiet" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case to keep quiet = hold ones tongue.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.