what prompts the argument between aunt Alexandra and Atticus?

That would depend on the chapter. My first thought goes to Alexandra's comment in regard to Calpurnia's role in the family, and the fact that she alludes they could do without her. This comment that truly fuels Atticus' anger, as he sees Calpurnia as one of the family.... as do the children. Needless to say, that subject was quickly closed.