The most general word for "argue" in Irish is "argóint", which more or less corresponds to English "arguing":
Argóint le duine faoi rud = To argue with someone about something
"Bhí mé ag argóint leis faoin áit ar cheannaigh sí an leabhar"
Another word you could use, perhaps if the argument is a bit more intellectual, is "díospóireacht", which can also mean "discussing, debating".
"Bhí mé ag díospóireacht leis faoi cé acu scannán a bhféachfaimis air anocht"
Yet another word is "sáraíocht" = "contending, disputing, arguing", which comes from the verb "sáraigh" = "to overcome, surpass". It gives an idea of each party in turn getting one up on the other, but it's used slightly differently than the other two words:
"Bhí siad ag sáraíocht ar a chéile faoi rud".