English Translation
They were constantly quarreling.
絶えず and ばかり mean the same and they are redundant.
彼らは喧嘩（けんか）ばかりしていた。
彼らは喧嘩が絶えなかった。（けんかがたえなかった）
are enough and concise.
～ばかりしている/いた means “to do something so many times” or “to do it repeatedly”. 絶える as an intransitive verb means “to be extinct”, and 絶えず as an adjective means “constantly” or “eternally”.