A terahertz (THz) heterodyne spectrometer is demonstrated based on a quantum cascade laser (QCL) as a local oscillator (LO) and an NbN hot electron bolometer as a mixer, and it is used to measure high-resolution molecular spectral lines of methanol (CH3OH) between 2.913–2.918 THz. The spectral lines are taken from a gas cell containing methanol gas and using a single-mode QCL at 2.9156 THz as an LO, which is operated in the free running mode. By increasing the pressure of the gas, line broadening and saturation are observed. The measured spectra showed good agreement with a theoretical model.