In this work, magnetoresistance of as-grown and annealed n- and p-type modulation-doped Ga0.68In0.32NyAs1-y/GaAs single quantum well structures with various nitrogen concentrations has been studied. At low temperatures and low magnetic fields, in n-type samples negative and in p-type samples positive, magnetoresistance has been observed. The observed negative magnetoresistance in n-type samples is an indication of enhanced backscattering of electrons due to the weak localization of the electrons as an effect of the N-induced defects. Nitrogen concentration and thermal annealing dependence of the magnetoresistance have been studied for both n- and p-type samples. The observed decrease in the negative magnetoresistance in n-type and enhanced positive magnetoresistance in p-type samples following thermal annealing have been explained by considering thermal annealing-induced improvement of mobility and the crystal quality in N-containing samples. After thermal annealing, the magnitude of negative magnetoresistance decreases and the breaking of the weak localization is achieved at lower magnetic fields in n-type samples. It is observed that as the mobility of the sample increases, critical magnetic field of negative to positive magnetoresistance transition becomes lower.