Proton therapy based on the linear accelerator is an advanced radiation treatment technology. Lanzhou University has proposed a linac scheme for proton therapy, which features a 750 MHz Radio Frequency Quadrupole (RFQ) as its first accelerator. The RFQ is capable of accelerating a 1 mA proton beam from 30 keV to 3 MeV. The article systematically introduces the beam dynamics design and error analysis of the RFQ. In the design part, a new strategy that can reduce the outlet longitudinal emittance of the RFQ by adjusting its modulation and synchronous phase was proposed. Subsequently, multi-particle simulations based on different codes consistently demonstrated that the dynamics design achieved a good output beam quality. The longitudinal emittance of the RFQ output beam was successfully controlled to below 0.1 π mm·mrad. In the error analysis part, firstly, all types of errors were analyzed separately to determine their impact patterns and to give preliminary tolerances. Then, the combined errors which considered all types of errors together were analyzed. Finally, the tolerances for input beam errors, cavity voltage errors, and mechanical errors were obtained.