A very high resolution (R = λ/Δλ = 200, 000), high signal-to-noise ratio (S/N ~= 340) blue-green spectrum of the very metal-poor ([Fe/H] ~= -2.6) red giant star HD 122563 has been obtained by us at McDonald Observatory. We measure the asymmetries and core wavelengths of a set of unblended Fe I lines covering a wide range of line strength. Line bisectors exhibit the characteristic C-shape signature of surface convection (granulation) and they span from about 100 m s-1 in the strongest Fe I features to 800 m s-1 in the weakest ones. Core wavelength shifts range from about -100 to -900 m s-1, depending on line strength. In general, larger blueshifts are observed in weaker lines, but there is increasing scatter with increasing residual flux. Assuming local thermodynamic equilibrium (LTE), we synthesize the same set of spectral lines using a state-of-the-art three-dimensional (3D) hydrodynamic simulation for a stellar atmosphere of fundamental parameters similar to those of HD 122563. We find good agreement between model predictions and observations. This allows us to infer an absolute zero point for the line shifts and radial velocity. Moreover, it indicates that the structure and dynamics of the simulation are realistic, thus providing support to previous claims of large 3D-LTE corrections to elemental abundances and fundamental parameters of very metal-poor red giant stars obtained with standard 1D-LTE spectroscopic analyses, as suggested by the hydrodynamic model used here.