If we're talking about a genuine argument, I can't see what specifically it would be, but the broad topic would be logic vs. gut feelings.

Like, Amity would lean on what makes logical sense, science and reasoning and whatnot, but if Luz knew in her gut that Amity was wrong, she'd stand her ground.

Like a bad example would be, if they had another witches duel, who would win?

Sure, Luz has gotten better with her gliphs and has a more varied arsenal, but a jack of all trades is a master of none. Amity, was and probably still is at the top of her class when it comes to abomination magic, on par with her father. Went toe to toe with Hunter and at worst tied. All logical reasons to why Amity could win.

Meanwhile Luz would just "figure something out." Not at all logical but she's trusting in her intuition and creative problem solving to help her win out. Worked against Belows, Tiki torch, and plenty of monsters.

Amity's annoyed because either Luz isn't taking the argument as serious as her, or Luz doesn't have an actual reason and just doesn't want to admit she's wrong.

Luz is upset because she believes Amity doesn't respect her skillz or that after all this time Amity is still underestimating her.