I think the velocity scale is calculated something like this:
$$v_r = c\left( \frac{\lambda - \lambda_0}{\lambda_0}\right),$$
where $\lambda$ is the observed wavelength and $\lambda_0$ is wavelength of the Mg II line, corrected for the redshift of the quasar ($z=0.868)$ it is observed towards.
The confusing thing is, for Fig.~3 in the cited paper, that the Mg II feature is actually doublet with lines at rest wavelengths of $\lambda_r =2796.35$ and $2803.53$ Angstroms. Therefore
$$\lambda_0 = \lambda_r (1+z) = 5223.58\ \ {\rm or} \ \ 5236.99\ \ {\rm Angstroms}$$
corresponding to the thick and thin solid lines on Fig.3 respectively.
In other words, zero on the velocity scale would correspond to a Mg II absorber at the redshift of the quasar (an intrinsic absorber), whereas negative velocities correspond to absorbers at lower redshift, by the velocities indicated.
For a check on this - calculate the velocity of feature "b", which appears to be at about 5221 Angstroms in the observed spectrum for the redder doublet component (the thin solid line in Fig.3). Using the above definition, this would be a velocity of -915 km/s.
Looking at Fig.3, there seems to be a major problem. Feature "b" is at -680 km/s. However, if we look further on at Figs. 4 and 5, we see feature "b" plotted at -850 km/s in the Mg II spectrum, which is much closer, and would perhaps be in exact agreement if feature "b" was actually at 5222 Angstroms (which looks unlikely) or that a 4th, unspecified decimal place had been used in the redshift. e.g. if $z=0.8676$ then $\lambda_0 = 5235.87$ Angstroms, and the velocity of a feature measured at 5221 Angstroms would then be -852 km/s in perfect agreement with Figs. 4 and 5.
Either way, there is major disagreement between Fig. 3 (which I suspect is in error) and Figs. 4 and 5 (which could be reconciled with the above analysis if the redshift was 0.8676).
I'm sorry this is not very conclusive and I certainly wouldn't accept it without someone else verifying that the x-axis of Fig.3 appears to be problematic.
