There are several versions of quantum game theory. First, there are Eisert-Wilkens type models in which players submit strategies that are quantum superpositions of classical strategies. In these models, a classical game {\bf G} is replaced by a quantum game {\bf G}, which is itself a classical game with a much larger strategy space than {\bf G}. So here the answerr to your question is: Yes, the "quantum game" is itself in fact a classical game.

Next, there are models in which players use quantum technology to randomize among (classical) strategies, where these technologies might include observations of entangled particles. As long as players have no private information, this is equivalent to allowing players to choose their strategies contingent on classical correlated random variables. (E.g. perhaps you and I each choose our strategies contingent on whether we see the sun shining; whether or not the sun shines near my house is correlated with whether or not it shines near yours.) So here all we've done by adding quantum strategies is imbed our classical game {\bf G} into a larger (classical) game. Every equilibrium in the expanded game is a correlated equilibrium in the original game (but not, in general, vice versa). Here again, the answer to your question is that a "quantum game" is nothing but a particular classical game.

However, genuinely new phenomena occur when players have access to private information, e.g. when they have only partial information about the payoff matrix (and one player's information might differ from the other's). Here each player has a set of basic strategies (say "Confess (C)" and "Not Confess" (N)), and a piece of information $x$ chosen from some information set $P$. We can then model a (classical) mixed strategy as either

a) A map from $P$ to probability distributions over $\lbrace N,C\rbrace $ or

b) A probability distribution over maps from $P$ to $\lbrace N, C\rbrace $

It turns out not to matter which model you choose; the two models (usually called "the game of behavioral strategies" and "the game of mixed strategies") are equivalent in an appropriate sense and make all the same predictions about payoffs. Whichever model you choose, you've got a new (classical) game to study.

But in the quantum environment, the equivalence between mixed and behavioral strategies breaks down, and genuinely new phenomena emerge. In this case, the quantum version of the behavioral-strategy game has equilibria that cannot be recognized as correlated equilibria in any naturally defined classical game. So in this case, the answer to your question becomes no.

In particular: If we allow players to choose a map from the information set $P$ to the set of quantum strategies over $\lbrace N,C\rbrace$ --- where a ``quantum strategy'' is a strategy contingent on the realization of a quantum observable --- and if the observables available to the two players are entangled --- then one gets equilibria that are not in any sense equivalent to correlated equilibria in the original classical game.

This is all spelled out in careful detail in Dahl and Landsburg, to which Carlo has already referred. But since the key diagrams are missing from that arxiv'd version (due to insane policies on the part of the arxiv), you might prefer to read it here.