  We show that the mutually exclusive nature of classical and quantum
correlations distributed in multi-party quantum systems can be characterized in
terms of $q$-expectation. Using Tsallis-$q$ entropy and $q$-expectation, we
first provide generalized definitions of classical and quantum correlations,
and establish their trade-off relations in three-party quantum systems of
arbitrary dimension with respect to $q$-expectation for $q\geq 1$. We also
provide equivalence conditions for monogamy and polygamy inequalities of
quantum entanglement and quantum discord distributed in three-party quantum
systems of arbitrary dimension with respect to $q$-expectation for $q\geq 1$.
