Atoms can be prepared at low temperatures by laser cooling. The center of mass motion of the cold atoms exhibits wave nature of quantum mechanics. In this paper, we study the quantum tunneling effect of a cold atomic beam propagating through a laser beam which is blue detuned. The blue detuned laser beam is physically equal to a quantum potential barrier. According to two-level model of atom, we systematically analyze the quantum reflection and transmission of the vector atomic matter wave with internal structures propagating through a laser beam. Quantum tunneling time the atomic beam spends in transmitting the laser beam is studied in details. Because of quantum wave nature, cold atoms tunneling through a laser beam can obviously have a completely different feature from that of a classical particle (hot atom).