Shut up in quiet part of hospital (4)
I believe the answer is:
pent
'shut up' is the definition.
(synonyms)
'quiet part of hospital' is the wordplay.
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'part of hospital' becomes 'ent' (abbreviation for Ear, Nose and Throat department).
'p'+'ent'='PENT'
'in' is the link.
(Other definitions for pent that I've seen before include "enclosed" , "Closely confined" .)