  In this paper we apply our new separation of variables approach to completely
characterize the transfer matrix spectrum for quantum integrable lattice models
associated to fundamental evaluation representations of $\mathcal{U}_{q}
(\widehat{gl_{n}})$ with general quasi-periodic boundary conditions. We
consider here the case of generic deformations associated to a parameter $q$
which is not a root of unity. The Separation of Variables (SoV) basis for the
transfer matrix spectral problem is generated by using the action of the
transfer matrix itself on a generic co-vector of the Hilbert space, following
the general procedure described in our paper [1]. Such a SoV construction
allows to prove that for general values of the parameters defining the model
the transfer matrix is diagonalizable and with simple spectrum for any twist
matrix which is also diagonalizable with simple spectrum. Then, using together
the knowledge of such a SoV basis and of the fusion relations satisfied by the
hierarchy of transfer matrices, we derive a complete characterization of the
transfer matrix eigenvalues and eigenvectors as solutions of a system of
polynomial equations of order $n+1$. Moreover, we show that such a SoV discrete
spectrum characterization is equivalently reformulated in terms of a finite
difference functional equation, the quantum spectral curve equation, under a
proper choice of the set of its solutions. A construction of the associated
Q-operator induced by our SoV approach is also presented.
