Using $(10.087\pm 0.044)\times {10}^9$ $J/\psi$ events and $(2.712\pm 0.014)\times {10}^9$ $\psi (3686)$ events collected by the BESIII detector operating at the BEPCII collider, we report the first observation of the hadronic decays of $J/\psi \to p\bar{p}{K}_S^0{K}_S^0$ and $\psi (3686)\to p\bar{p}{K}_S^0{K}_S^0$, both with statistical significance greater than $10\sigma$. Their branching fractions are determined to be $ℬ(J/\psi \to p\bar{p}{K}_S^0{K}_S^0)=(1.60\pm 0.02\pm 0.09)\times {10}^{-5}$ and $ℬ(\psi (3686)\to p\bar{p}{K}_S^0{K}_S^0)=(3.93\pm 0.24\pm 0.34)\times {10}^{-6}$. The ratio of their branching fractions is $ℬ(\psi (3686)\to p\bar{p}{K}_S^0{K}_S^0)/ℬ(J/\psi \to p\bar{p}{K}_S^0{K}_S^0)=(24.6\pm 1.5\pm 2.1)\%$, which deviates from theoretical expectation by 4.6$\sigma$. Here the first uncertainties are statistical and the second systematic. We have also examined the $p\bar{p}$ invariant mass distributions in these decays, and no significant enhancement near the $p\bar{p}$ near threshold is found.