  PG 1553+113 is a well-known blazar exhibiting evidence of a $\sim\! 2.2$-yr
quasi-periodic oscillation (QPO) in radio, optical, X-ray, and $\gamma$-ray
bands. Since QPO mechanisms often predict multiple QPOs, we search for a second
QPO in its historical optical light curve covering a century of observations.
Despite challenging data quality issues, we find hints of a $21.8 \pm 4.7$ yr
oscillation. On its own, this $\sim\! 22$-yr period has a modest statistical
significance of $1.6\sigma$ when accounting for the look-elsewhere effect.
However, the joint significance of both the $2.2$- and $22$-yr periods arising
from colored noise alone is $\sim 3.6\sigma$. The next peak of the 22-yr
oscillation is predicted to occur around July 2025. We find that such a
$\sim\,$10:1 relation between two periods can arise in the gas dynamics of a
plausible supermassive black hole binary model of PG 1553+113. While the 22-yr
QPO is preliminary, an interpretation of PG 1553+113's two QPOs in this binary
model suggests that the binary engine has a mass ratio $\gtrsim 0.2$, an
eccentricity $\lesssim 0.1$, and accretes from a disk with characteristic
aspect ratio $\sim 0.03$. The putative binary radiates nHz gravitational waves,
but the amplitude is $\sim10-100$ times too low for detection by foreseeable
pulsar timing arrays.
