Designate a quiet period (7)
I believe the answer is:
appoint
'designate' is the definition.
(thesaurus)
'a quiet period' is the wordplay.
'a quiet' becomes 'ap' (I've seen this in other clues).
'period' becomes 'point' (point is a kind of period**).
'ap'+'point'='APPOINT'
(Other definitions for appoint that I've seen before include "Assign job to (someone)" , "Put into a post" , "Pick" , "Assign to a duty" , "Assign position to" .)