Quarrelsomeness is a state of being quick to argue or disagree. The triplets' quarrelsomeness made it hard for their parents to find a babysitter who was willing to put up with them.
To quarrel means to bicker or squabble, from the Latin querella, "complaint or accusation." Quarrelsomeness is a tendency to quarrel, picking fights or finding fault. You might decide to stop watching the news on TV because of all the political quarrelsomeness. And your sister's quarrelsomeness makes it seem like she always manages to start an argument with you, even about silly things.