Can countries introduce new aviation policies unilaterally? Let's say some country intends to introduce a new aviation policy that conflicts with ICAO regulations, like only aircraft certified and approved by our government can fly in our airspace. Can they do that unilaterally? Has it ever happened?

For instance, Russia may declare a country as enemy country and it may use its equipment (e.g reconnaissance aircraft) to spy on this enemy country. In this case enemy country may start defining its own policies and make sure that defense or civil aviation equipment/software are used in compliance with these new policies.
Aircrafts need to be certified to operate under FAA/European/Russian... regulations in terms of safety, noise, etc.
For instance, in Europe EASA is the authority "for the airworthiness and environmental certification of all aeronautical products[..]".

Sometimes countries actually want to introduce new policies.
One of the most famous was in 1971, when "The US, India, and Malaysia all ruled out Concorde supersonic flights over the noise concern[..]" as reported here on Wikipedia.

Another one about aircraft noise was in 1999, when "The EU's proposed noise ordinances effectively prevented the use of hush-kit outfitted aircraft in Europe[..]".

So every country already has its own regulation.