Computer simulation is carried out for studying the microphase separation of a two-dimensional diblock copolymer (DBCP) system under directional quenching. By setting the quenching boundary between the stable and the unstable phase, and shifting the boundary with a constant velocity, the time evolution of the domain morphologies is examined numerically on the basis of the time-dependent Ginzburg-Landau type equation with the free-energy functional for the DBCP. Three different types of morphologies are found for the symmetric (i.e. f=0.5) DBCP system. One is the irregular lamellar morphology and is essentially equivalent to that produced by homogeneous quenching. The other two are regular and are characteristics of directional quenching process. One of the regular lamellar morphologies is perpendicular to the quench boundary on the average, whereas the other one is parallel to the quench boundary. For the asymmetric DBCP system with f=0.4, which forms the equilibrium morphology of triangular phase, the mode of regular lamellar morphology with the normal of lamellae perpendicular to the quench boundary appears first under the condition of directional quenching when the initial thermal fluctuation is very small. The growth rate of the triangular phase gets faster when the initial thermal fluctuation increases. Therefore, our results reveal that the thermal fluctuation promotes the appearance of the mode of triangular morphology for the case of f=0.4.