We theoretically analyze type-I broadband second-harmonic generation (SHG) of femtosecond laser pulses based on a quasi-phase-matching configuration in periodically poled congruent {\mathrm{LiNbO}}_3 (LN) and periodically poled \mathrm{MgO}:{\mathrm{LiNbO}}_3 (PPMgLN) (5% and 7%). Group-velocity matching (GVM) can be achieved at the fundamental waves of 1.59, 1.56, and 1.55\mathrm{\mu m} for SHG when the three types of crystals have grating periods of 22.31, 20.07, and 23.45\mathrm{\mu m} , respectively. It is found that the central wavelength of the fundamental wave for GVM will increase with the decrease of MgO doping in \mathrm{L}\mathrm{N} . It is concluded that the shift of the GVM central wavelength is due to the difference of MgO doping, which changes the dispersion of the crystal. Therefore, tunable and high efficiency broadband SHG of femtosecond laser pulses in a long crystal can be realized by selecting different doping rates of PPMgLN.