Argument after Conservative's boast (4)
I believe the answer is:
crow
'boast' is the definition.
(to crow about something is to boast about it)
'argument after conservative's' is the wordplay.
'argument' becomes 'row' (to row is to argue).
'after' is a charade indicator (letters next to each other).
'conservative' becomes 'C' (UK political abbreviation).
'row' after 'c' is 'CROW'.
(Other definitions for crow that I've seen before include "Black bird; boast" , "Black bird with raucous call" , "Bird - birdcall" , "'As the - - flies'" , "Express triumph; bird" .)