  We study the relation between two kinds of topological amplitudes of
non-compact D-branes on conifold. In the A-model, D-branes are represented by
fermion operators in the melting crystal picture and the amplitudes are given
by the quantum dilogarithm. In the mirror B-model, D-branes correspond to the
determinant operator det(x-M) in the Chern-Simons matrix model and the
amplitudes are given by the Stieltjes-Wigert polynomial. We show that these two
amplitudes are related by a certain integral transformation. We argue that this
transformation represents the deformation of closed string background due to
the presence of D-branes.
