"quarreled over" is a correct and usable phrase in written English.
It is usually used to describe a disagreement or conflict between two or more people, usually regarding a particular topic. For example: "John and his brother quarreled over who should get to keep the last piece of cake."
From the beginning, the state and the administration have quarreled over who should get most of the newly captured water -- the natural system, Florida's exploding cities, or agriculture.
The government has also quarreled over contract provisions with an American company, Science Applications International Corporation, which was hired to provide an elaborate security system to spot suspicious activity and detect chemical, biological and nuclear weapons.
Standing stiffly in a dark suit with his court-appointed public defender by his side, Father Fay, 56, said he generally agreed with the government's conclusions but quarreled over some details, including the amount he had stolen.
The top prosecutors of Russia and Britain quarreled over a British demand to extradite a former K.G.B. security officer accused of poisoning Alexander V. Litvinenko in London last November.
They have quarreled over his job creation record.
The candidates quarreled over subsidies for the oil industry, Medicare cuts, taxes and government spending.
The two sides quarreled over verifying compliance, sharing telemetry and limiting missile defense programs.
The French did the same with Syria and Lebanon while everyone quarreled over Palestine.
When we quarreled over the network's coverage of the Vietnam hearings in 1966, I quit.
Actually, debated is too tepid a word -- it should be disputed, quarreled over, with intelligence.
He remember their last meeting, and how they had quarreled over the menu.
