If NE is a CK? It seems that yes since given all information about payoffs/strategies players can derive NE based on the procedures similar to that of in the common knowledge, but I'm not sure.
If the full payoff matrix for everyone was part of the common knowledge, then of course everyone will be able to observe that a given position is a Nash equilibrium, by observing that no-one will be able to improve their position by changing individually, and this calculation would also be part of the common knowledge (provided it was common knowledge that everyone could perform this calculation). So in this case, it would be common knowledge whether a given position was a Nash equilibrium.
But if the payoff matrix was not part of the common knowledge, for example, if people only know their own payoffs, then it could be that they find themselves in a Nash equilibrium, in the sense that no-one can improve their payoff by changing only their own value, but not knowing sufficiently much of the full payoff matrix, they wouldn't know whether someone else could improve their position, and in this imperfect information situation being in a Nash equilibrium wouldn't be common knowledge.
Even if everyone knows the full payoff matrix, but doesn't know that everyone knows it (or perhaps they know everyone else knows it, but don't know that everyone else knows that everyone knows it), then it could be that everyone knows the current position is a Nash equilibrium, but it still might not be part of the common knowledge, because perhaps not everyone knows that everyone knows it.