There are many instances in history, though some of them depend on where you make the distinction between government and country. The French in World War II for example, started out with the Allies, was conquered by Germany, which installed the puppet Vichy regime. Forces loyal to this regime did fight against the Allies in several areas, including North Africa and Syria. They fought particularly hard against the Allies at Oran, where the British had sunk much of the French Fleet two years earlier, in an attempt to keep it out of German hands. However, throughout the war, organized Free French units fought on the side of the Allies, even landing with the American Seventh Army in southern France during operation dragoon. This isn't exactly an area I have studied extensively, just the most modern example I can remember.

My source on this is:

The Second World War, by Winston Churchill. It's from the volumes 2 3 and 4 of the condensed edition. I know this is an autobiographical source, but I feel it is sufficient for this rather basic example.

France in WWII is the only thing I can think of that comes close. You have Vichy France that cooperates with the Nazi regime, and you have the Free French movement who does not.

The Soviet Union started World War II as a co-belligerent with Germany in conquering Poland only to fight Gemany two years later.

In the Napoleonic Wars, Austria, Russia and Spain were on both sides at different times. The U.S. in the meantime fought the Quasi War against France and later the War of 1812 against Britain.

The Hundred Years War.

Various vassals of the French monarchs - as well as English kings who had a claim to the French throne - warred with their nominal liege lord (the King of France).

A Distant Mirror by Barbara Tuchmann goes into this in accessible detail.

Along with the France in WWII example, what about Korea in the Korean War or Vietnam in the Vietnam war?

There are plenty of examples of outside powers backing more than one side in civil wars and proxy conflicts during the Cold War, particularly ones with more than two sides. A few examples off the top of my head; the four-sided civil war in Cambodia between royalists, anti-Communists, the Khmer Rouge, and Vietnamese-aligned Communists, where the PRC/USA/USSR/Thailand/North Vietnam/South Vietnam were frequently backing more than one side at once (most of them backed every side at some point), or the five-sided civil war in Lebanon between Christians, Sunni Islamists, the PLO and far-leftists, Shia Islamists, and the Lebanese central government, where Israel/Syria/USA/USSR were also generally backing more than one side at once, or Angola, where South Africa was simultaneously backing the Portuguese colonial forces and the FNLA and UNITA.