Precision radial velocity (RV) measurements in the near-infrared are a powerful tool to detect and characterize exoplanets around low-mass stars or young stars with higher magnetic activity. However, the presence of strong telluric absorption lines and emission lines in the near infrared that significantly vary in time can prevent extraction of RV information from these spectra by classical techniques, which ignore or mask the telluric lines. We present a methodology and pipeline to derive precision RVs from near-infrared spectra using a forward-modeling technique. We applied this to spectra with a wide wavelength coverage (Y, J, and H bands, simultaneously), taken by the InfraRed Doppler (IRD) spectrograph on the Subaru 8.2-m telescope. Our pipeline extracts the instantaneous instrumental profile of the spectrograph for each spectral segment, based on a reference spectrum of the laser-frequency comb that is injected into the spectrograph simultaneously with the stellar light. These profiles are used to derive the intrinsic stellar template spectrum, which is free from instrumental broadening and telluric features, as well as model and fit individual observed spectra in the RV analysis. Implementing a series of numerical simulations using theoretical spectra that mimic IRD data, we test the pipeline and show that IRD can achieve $<2$ m s$^{-1}$ precision for slowly rotating mid-to-late M dwarfs with a signal-to-noise ratio $> 100$ per pixel at 1000 nm. Dependences of RV precision on various stellar parameters (e.g., $T_{rm eff}$, $vsin i$, [Fe/H]) and the impact of telluric-line blendings on the RV accuracy are discussed through the mock spectra analyses. We also apply the RV-analysis pipeline to the observed spectra of GJ 699 and TRAPPIST-1, demonstrating that the spectrograph and the pipeline are capable of an RV accuracy of $<3$ m s$^{-1}$ at least on a time scale of a few months.
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