University engaged in supporting a quartet (4)
I believe the answer is:
four
'a quartet' is the definition.
(I know that quartet can be written as four)
'university engaged in supporting' is the wordplay.
'university' becomes 'U' (abbreviation for university).
'engaged in' means one lot of letters goes inside another.
'supporting' becomes 'for' (being for something means supporting it).
'u' put into 'for' is 'FOUR'.
(Other definitions for four that I've seen before include "Set of bridge players?" , "Cube root of 64" , "Number of players in a doubles game" , "10 cleared of misuse of fortune" , "Figure" .)