Yes, just about every occupied country did have insurgencies in one form or another. Here is an example.
The most successful in Europe was the National Liberation Army and Partisan Detachments of Yugoslavia (later known as the Yugoslav army). It was the only country to self liberate from the Fascists without major foreign intervention.
On 6 April 1941, the Kingdom of Yugoslavia was invaded from all sides by the Axis powers, primarily by German forces but including Italian, Hungarian and Bulgarian formations as well. During the invasion, Belgrade was bombed by the Luftwaffe. The invasion lasted little more than ten days, ending with the unconditional surrender of the Royal Yugoslav Army on 17 April. Besides being hopelessly ill-equipped when compared to the Wehrmacht, the Army attempted to defend all borders but only managed to thinly spread the limited resources available.
The Partisan movement in Yugoslavia initially began as a socialist resistance led by the Communist Party of Yugoslavia, led by Josip Broz Tito, which began in 1941. In the country however there wasn't a unified resistance.
There were multiple factions in Yugoslavia, some axis and some allied, all competing for control. Some fought each other as much as they did the Fascist. The partisan resistance movement led by Tito eventually absorbed several other resistance movements forming a unified anti-fascist movement, however the heads of the resistance were still mostly communists.
The Partisans staged a guerrilla campaign which enjoyed gradually increased levels of success and support of the general populace, and succeeded in controlling large chunks of Yugoslav territory. These were managed via the People's committees, which were organized to act as civilian governments in areas of the country controlled by the communists, even limited arms industries were set up.
At the very beginning, though, Partisan forces were relatively small, poorly armed and without any infrastructure. But they had two major advantages over other military and paramilitary formations in former Yugoslavia: the first and most immediate was a small but valuable cadre of Spanish Civil War veterans who, unlike anyone else at the time, had experience with modern war fought in circumstances quite similar to those of World War II Yugoslavia.
Another advantage, which became apparent in later stages of war, was in Partisans being founded on ideology rather than ethnicity, which meant the Partisans could expect at least some levels of support in any corner of the country, unlike other paramilitary formations whose support was limited to territories with Croat or Serb majorities. This allowed their units to be more mobile and fill their ranks with a larger pool of potential recruits. They were a multi-ethnic, mutual-religious army.
While you're waiting for experts to answer, you might be interested in reading some previous threads on partisan activity in Nazi-occupied Europe:
Insurgencies
Nazi Counter-Insurgency