  We present a pipeline to identify photometric variability within strong
gravitationally lensing candidates, in the DESI Legacy Imaging Surveys. In our
first paper (Sheu et al. 2023), we laid out our pipeline and presented seven
new gravitationally lensed supernovae candidates in a retrospective search. In
this companion paper, we apply a modified version of that pipeline to search
for gravitationally lensed quasars. From a sample of 5807 strong lenses, we
have identified 13 new gravitationally lensed quasar candidates (three of them
quadruply-lensed). We note that our methodology differs from most lensed quasar
search algorithms that solely rely on the morphology, location, and color of
the candidate systems. By also taking into account the temporal photometric
variability of the posited lensed images in our search via difference imaging,
we have discovered new lensed quasar candidates. While variability searches
using difference imaging algorithms have been done in the past, they are
typically preformed over vast swathes of sky, whereas we specifically target
strong gravitationally lensed candidates. We also have applied our pipeline to
655 known gravitationally lensed quasar candidates from past lensed quasar
searches, of which we identify 13 that display significant variability (one of
them quadruply-lensed). This pipeline demonstrates a promising search strategy
to discover gravitationally lensed quasars in other existing and upcoming
surveys.
