  We propose a natural generalization of the construction of the quantum
difference Toda lattice (introduced independently by Etingof and Sevostyanov)
associated to a simple Lie algebra $\mathfrak{g}$. Our construction depends on
two orientations of the Dynkin diagram of $\mathfrak{g}$ and some other data
(which we refer to as a pair of Sevostyanov triples). In types $A$ and $C$, we
provide an alternative construction via Lax matrix formalism, generalizing the
one of Kuznetsov-Tsyganov for the classical $q$-Toda. We also show that the
generating function of the pairing of Whittaker vectors in the Verma modules is
an eigenfunction of the corresponding modified quantum difference Toda system
and derive fermionic formulas for the former in spirit of the work by
Feigin-Feigin-Jimbo-Miwa-Mukhin. We give a geometric interpretation of all
Whittaker vectors in type $A$ via line bundles on the Laumon moduli spaces and
obtain an edge-weight path model for them, slightly generalizing the
construction of Di Francesco-Kedem-Turmunkh.
