Thus, on the quiet, it's a peace offering (3)
I believe the answer is:
sop
'a peace offering' is the definition.
(I've seen this in another clue)
'thus on the quiet' is the wordplay.
'thus' becomes 'so' (synonyms).
'on' says to put letters next to each other (in a down clue, letters appear on others).
'the quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'so'+'p'='SOP'
'it's' acts as a link.
(Other definitions for sop that I've seen before include "Bread dipped in gravy" , "Minor concession" , "Small concession" , "Something of no great value given to appease" , "usual way of doing things, initially" .)