It really depends on the time period (the US went through several distinct "phases" of Civil Defense training and preparation), but essentially, the answer is yes. The Soviet Union had entire high school courses dedicated to Civil Defense (and continued these well into the Russian Federation period, though the students have long regarded them as a joke). Drills on Civil Defense took place in many other nations that considered themselves a possible targets in a nuclear attack, notably the UK, even though many of its officials concluded that the UK would essentially not be able to weather a full, megaton-range attack under any circumstances, and West Germany.

The difficulty in some of this is that the notion of the USSR having "advanced" Civil Defense preparation, and having far more facilities and resources than it did invest in the matter, became a trope of the right-wing in the United States in the 1970s and 1980s, a reason to argue that the USSR was ready and prepared for a first-strike attack against the USA, and thus a justification for an arms build-up. So the truth of such claims can be controversial.

It is also of note that "Duck and Cover" is indicative of a very specific form of Civil Defense (mitigation of the blast effect) that was common in the pre-megaton age. Once thermonuclear weapons were deployed Civil Defense policies were more concerned with fallout and possible evacuation measures (because the blast effects of a megaton weapon are hard to mitigate unless you are really on the fringes of the detonation).