Topological vertex is a diagrammatic method for constructing the partition functions (or amplitudes) of topological string theory on non-compact toric Calabi-Yau threefolds. This talk is focussed on two cases where open string amplitudes can be computed explicitly by this method. One is the case of ``on-strip'' geometry, and the other is the so called ``closed topological vertex''. For both cases, generating functions of open string amplitudes turn out to satisfy a linear q-difference equation. This equation may be thought of as quantization of the equation of the mirror curve.