We analyze theoretically nonlinear dynamics of an optically injected two-mode quantum dot laser lasing simultaneously from the ground and excited states. We show that although the external optical signal is injected into the ground-state mode alone, it can lead to the generation of regular picosecond pulses and pulse packages in the intensity of the excited-state mode. Generation of regular streams of picosecond pulses is attributed to an intrinsic gain switching mechanism where the relaxation time is modulated by the oscillations in the occupation of the ground and excited energy states