  It was known that the partition functions of the U(N) x U(N+M) ABJM theory
satisfy a set of bilinear relations, which, written in the grand partition
function, was recently found to be the q-Painleve III_3 equation. In this paper
we have suggested a similar bilinear relation holds for the ABJM theory with
N=6 preserving mass deformation for an arbitrary complex value of mass
parameter, to which we have provided several non-trivial checks by using the
exact values of the partition functions for various N,k,M and the mass
parameter. For particular choices of the mass parameters labeled by integers
$\nu,a$ as $m_1=m_2=-\pi i(\nu-2a)/\nu$, the bilinear relation corresponds to
the q-deformation of the affine SU($\nu$) Toda equation in $\tau$-form.
