  We report, for the first time, a synchronized velocity shift of Mg II and Al
III broad absorption lines (BALs) in quasar SDSS J134444.33+315007.6 (hereafter
J1344+3150). We found this quasar from a sample of 134 Mg II BAL quasars with
multi-epoch observations. This quasar contains three low-ionization BAL
systems, the fastest of which at $\sim$$-17, 000\,\rm km\,s^{-1}$ shows a
kinematic shift of $\sim$$-1101\,\rm km\,s^{-1}$ and $\sim$$-1170\,\rm
km\,s^{-1}$ in its Mg II and Al III ions, respectively, during a rest-frame
time of about 3.21 yr. Meanwhile, this quasar also shows other various
variation characteristics, including an obvious weakening in its continuum, a
coordinated enhancement in multiple emission lines (Mg II, C III and Al III),
and a coordinated enhancement in three Al III absorption troughs. These
variation characteristics convincingly indicate that the BAL outflows of
J1344+3150 are under the influence from the background radiation energy. Thus,
we infer that the velocity shift displayed in system A in the quasar J1344+3150
may indicate an actual line-of-sight acceleration of an outflow due to the
radiation pressure from the central source.
