We propose a generalization of the quantum relative entropy by considering the geodesic on a manifold formed by all the invertible density matrices \begin{document}$ \mathcal{P} $\end{document} . This geodesic is defined from a deformed exponential function \begin{document}$ \varphi $\end{document} which allows to work with a wider class of families of probability distributions. Such choice allows important flexibility in the statistical model. We show and discuss some properties of this proposed generalized quantum relative entropy.