Quiet after Greek character digs for fungi (9)
I believe the answer is:
mushrooms
'fungi' is the definition.
(mushroom is a kind of fungus)
'quiet after greek character digs' is the wordplay.
'quiet' becomes 'sh'.
'after' says to put letters next to each other.
'greek character' becomes 'mu' (letter from the Greek alphabet).
'digs' becomes 'rooms' ('digs' can mean a person's lodgings).
'sh' after 'mu' is 'mush'.
'mush'+'rooms'='MUSHROOMS'
'for' is the link.
(Other definitions for mushrooms that I've seen before include "Grows rapidly" , "undergoes a boom" , "Develops rapidly" , "builds up alarmingly" , "Fleshy fungi" .)