A quarrel about petty points is a disagreement between two or more people over trivial matters. In this case, the crossword solutions for the term are SPAT (4 letters), TIFF (4 letters), BICKER (6 letters), and NITS (4 letters). A SPAT is a brief argument, a TIFF is a minor dispute, BICKER is to argue in a petty way, and nits are small details. Therefore, a quarrel about petty points is an argument over small, insignificant matters.