Nick quiet: quiet sitting in cage (3,4)
I believe the answer is:
cop shop
'nick' is the definition.
(both are slang for police station)
'quiet quiet sitting in cage' is the wordplay.
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'quiet' becomes 'sh' (both can be said to tell someone to be quiet).
'sitting in' indicates putting letters inside.
'cage' becomes 'coop' (synonyms).
'p'+'sh'='psh'
'psh' inserted inside 'coop' is 'COP-SHOP'.
(Other definitions for cop shop that I've seen before include "Police station (slang)" , "Nick" , "'Police station (3,4)'" , "office investigating crime" .)