In this paper, we report our novel Monte Carlo quasi self-consistent particle simulation method for both electron and phonon transport in nanometer-sized electron devices. We developed two kinds of simulation procedures for the Monte Carlo method. First, we made a program to estimate the local temperature from a phonon spatial distribution, where we used a Bose-Einstein distribution function, the phonon density of states, and the phonon generation rate. Second, we developed an algorithm that made it possible to calculate multiple time scale phenomena of electron and phonon transport by introducing different time steps for electron and phonon transport simulations. With these methods, we succeeded in executing quasi self-consistent simulations of both electron and phonon transport in nanometer-channel FETs in consideration of saving computer processing time. Using these methods, we simulated the local heating properties of nanometer-scale gallium nitride FETs for the first time. Our FET model includes highly doped source and drain regions near the electrodes. It was found that phonon generation takes place mainly in the highly doped drain region, rather than in the high electric field regions of the channel or between the gate and drain. We discuss the physical basis of the spatial distributions of heat generation and local temperature in the GaN channel.