  We report the discovery of new lensed quasar candidates in the imaging data
of the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) DR4, covering
$1\,310~{\rm deg}^2$ of the sky with seeing of $\approx0.6''$. In addition to
two catalogs of MILLIQUAS and AllWISEAGN, which contain confirmed and candidate
quasars, we preselect quasar sources using color cuts from the HSC ($grizy$)
and unWISE ($W1+W2$) photometric data based on SDSS spectroscopic catalogs. We
search for the presence of multiple point sources with similar color through
the convolution of the Laplacian of the preselected quasar image cutouts with
the Laplacian of the point spread function, resulting in a reduction of lens
candidates from $1\,652\,329$ to $121\,511$ ($7.4\%$). After visual binary
classification, we grade $6\,199$ ($0.4\%$) potential lenses on a scale of 0 to
3, with 3 indicating a lens and 0 indicating a non-lens. Finally we obtain 162
lens candidates with an average grade of $\geq2$, and among them, we
successfully recover 18 known lenses. By fitting the light distribution and
removing the known contaminants, we discover that 57 new systems contain at
least two point sources and a galaxy in between, including 10 possible
quadruply lensed quasars. This new sample of lens candidates exhibits a median
separation of $1.26''$ and a magnitude limit of $i\approx22$. Spectroscopic or
high-resolution imaging follow up on these newly discovered lensed quasar
candidates will further allow their natures to be confirmed.
