Quickly getting close in public (9)
I believe the answer is:
overnight
'quickly' is the definition.
(eg something might change overnight)
'close in public' is the wordplay.
'close' becomes 'nigh' (similar in meaning).
'in' is an insertion indicator.
'public' becomes 'overt' (both can mean well-known or unconcealed).
'nigh' placed within 'overt' is 'OVERNIGHT'.
'getting' is the link.
(Other definitions for overnight that I've seen before include "from one day to the next" , "Lightning" , "Very quickly" , "Until morning; suddenly" , "darkness" .)