We determine Mg abundances in six Gaia benchmark stars using theoretical one-dimensional (1D) hydrostatic model atmospheres, as well as temporally and spatially averaged three-dimensional (〈3D〉) model atmospheres. The stars cover a range of Teff from 4700 to 6500 K, log g from 1.6 to 4.4 dex, and [Fe H] from -3.0 dex to solar. Spectrum synthesis calculations are performed in local thermodynamic equilibrium (LTE) and in non-LTE (NLTE) using the oscillator strengths recently published by Pehlivan Rhodin et al. We find that: (a) Mg abundances determined from the infrared spectra are as accurate as the optical diagnostics, (b) the NLTE effects on Mg I line strengths and abundances in this sample of stars are minor (although for a few Mg I lines the NLTE effects on abundance exceed 0.6 dex in 〈3D〉 and 0.1 dex in 1D, (c) the solar Mg abundance is 7.56 ± 0.05 dex (total error), in excellent agreement with the Mg abundance measured in CI chondritic meteorites, (d) the 1D NLTE and 〈3D〉 NLTE approaches can be used with confidence to analyze optical Mg I lines in spectra of dwarfs and sub-giants, but for red giants the Mg I 5711 line should be preferred, (e) low-excitation Mg I lines are sensitive to the atmospheric structure; for these lines, LTE calculations with 〈3D〉 models lead to significant systematic abundance errors. The methods developed in this work will be used to study Mg abundances of a large sample of stars in the next paper in the series.