  This paper explores integrable structures of a generalized melting crystal
model that has two $q$-parameters $q_1,q_2$. This model, like the ordinary one
with a single $q$-parameter, is formulated as a model of random plane
partitions (or, equivalently, random 3D Young diagrams). The Boltzmann weight
contains an infinite number of external potentials that depend on the shape of
the diagonal slice of plane partitions. The partition function is thereby a
function of an infinite number of coupling constants $t_1,t_2,...$ and an extra
one $Q$. There is a compact expression of this partition function in the
language of a 2D complex free fermion system, from which one can see the
presence of a quantum torus algebra behind this model. The partition function
turns out to be a tau function (times a simple factor) of two integrable
structures simultaneously. The first integrable structure is the bigraded Toda
hierarchy, which determine the dependence on $t_1,t_2,...$. This integrable
structure emerges when the $q$-parameters $q_1,q_2$ take special values. The
second integrable structure is a $q$-difference analogue of the 1D Toda
equation. The partition function satisfies this $q$-difference equation with
respect to $Q$. Unlike the bigraded Toda hierarchy, this integrable structure
exists for any values of $q_1,q_2$.
