A simple way of suppression of self-mode-locking in a nanosecond Q-switched Nd3+:YAlO3 laser by placing an element introducing a negative feedback into the laser cavity, which consists of a plate of singlecrystal GaAs exhibiting two-photon absorption (complete suppression) or a cell containing colloidal solution of CdSe/ZnS quantum dots (partial suppression), is implemented. Placing the element introducing the negative feedback into the cavity of a pulsed picosecond mode-locked Nd3+:Y3Al5O12 laser allowed an increase in the number of pulses in the pulse train and a change in the energy distribution between the pulses. Specificities of laser oscillation regimes in the presence of a nonlinear absorbing element in the cavity were analyzed by numerically solving the set of balance equations describing the population inversion density and the photon flux density in the cavity.