Periodically variable quasars have been suggested as close binary supermassive black holes. We present a systematic search for periodic light curves in 625 spectroscopically confirmed quasars with a median redshift of 1.8 in a 4.6 deg$^2$ overlapping region of the Dark Energy Survey Supernova (DES-SN) fields and the Sloan Digital Sky Survey Stripe 82 (SDSS-S82). Our sample has a unique 20-year long multi-color ($griz$) light curve enabled by combining DES-SN Y6 observations with archival SDSS-S82 data. The deep imaging allows us to search for periodic light curves in less luminous quasars (down to $r{sim}$23.5 mag) powered by less massive black holes (with masses $gtrsim10^{8.5}M_{odot}$) at high redshift for the first time. We find five candidates with significant (at $>$99.74% single-frequency significance in at least two bands with a global p-value of $sim$7$times10^{-4}$--3$times10^{-3}$ accounting for the look-elsewhere effect) periodicity with observed periods of $sim$3--5 years (i.e., 1--2 years in rest frame) having $sim$4--6 cycles spanned by the observations. If all five candidates are periodically variable quasars, this translates into a detection rate of ${sim}0.8^{+0.5}_{-0.3}$% or ${sim}1.1^{+0.7}_{-0.5}$ quasar per deg$^2$. Our detection rate is 4--80 times larger than those found by previous searches using shallower surveys over larger areas. This discrepancy is likely caused by differences in the quasar populations probed and the survey data qualities. We discuss implications on the future direct detection of low-frequency gravitational waves. Continued photometric monitoring will further assess the robustness and characteristics of these candidate periodic quasars to determine their physical origins.
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