It is not impossible to answer this. It is very clear, yes, spat means argument in this case. "Spat over" is not the phrase here. They are separate. For example: Prime Minister wants to rebuild ties with China in the wake of the spat. This argument was over last fall's maritime collisions near the disputed Senkaku Islands"

spat = argument 対立 They had a spat (argument) over something. "over" is the correct preposition to use with "spat"

It's impossible to answer this question without more information, specifically the use of the word in a sentence. "Spat" could be a noun meaning a small argument or it could be the past tense of the verb "spit." If it is used an a sentence like, "They had a little spat over some meaningless problem," then it means a petty argument.