This claim depends on your definition of a tax increase. As governor of Massachusetts, elected in 2002, Romney inherited a $3-billion state budget shortfall. And it's true that the budget problem was fixed without raising state income taxes. But Romney closed loopholes in the corporate income tax, which effectively increased taxes for some companies. And he and the legislature did increase a myriad of fees, on such things as boat registrations and court filings, which some might consider tax increases.
Also, state payments to cities and towns for schools and police were reduced, which caused those local governments to increase property taxes.
The Club for Growth, a conservative political action committee that pushes for spending cuts and tax cuts, evaluated Romney's fiscal record: "Overall, Romney's record on tax policy is mixed. His record is marred by questionable statements and positions and his fee hikes and 'loophole' closures are troubling."
We will note that the Club for Growth applauded Romney for supporting broad-based tax cuts.
We find his claim that he "cut spending instead of raising taxes" to be only half of the story about how he dealt with the state's financial crisis.