Conservative to quarrel and brag (4)
I believe the answer is:
crow
'brag' is the definition.
(I know that brag can be written as crow)
'conservative to quarrel' is the wordplay.
'conservative' becomes 'C' (UK political abbreviation).
'to quarrel' becomes 'row' (synonyms).
'c'+'row'='CROW'
'and' is the link.
(Other definitions for crow that I've seen before include "one often getting stoned!" , "Boast gloatingly, American Indian" , "Bird - birdcall" , "Blow one's own trumpet" , "Boast, swagger, triumph" .)