When referring to dictionaries, there seems to be no such meaning as “quarrel” under the word “square”, only “in agreement”.
But in II 1 of A Midsummer Night’s Dream, “square” in the following text means “quarrel”:
And now they never meet in grove, or green,
By fountain clear, or spangled starlight sheen,
But they do square, that all their elves for fear
Creep into acorn cups, and hide them there.
Where does this meaning come from?
The sense of to “fall out, to be at variance or discord, to disagree or quarrel” (OED, sense 8b of square, v.; paywalled) is obsolete. It appears in historical dictionaries. OED attests the sense with quotes from 1542 to 1608.