Using data samples of $102times10^6$ $Upsilon(1S)$ and $158times10^6$ $Upsilon(2S)$ events collected with the Belle detector, a first experimental search has been made for double-charmonium production in the exclusive decays $Upsilon(1S,2S)rightarrow J/psi(psi)+X$, where $X=eta_c$, $chi_{cJ} (J=~0,~1,~2)$, $eta_c(2S)$, $X(3940)$, and $X(4160)$. No significant signal is observed in the spectra of the mass recoiling against the reconstructed $J/psi$ or $psi$ except for the evidence of $chi_{c1}$ production with a significance of $4.6sigma$ for $Upsilon(1S)rightarrow J/psi+chi_{c1}$. The measured branching fraction $BR(Upsilon(1S)rightarrow J/psi+chi_{c1})$ is $(3.90pm1.21(rm stat.)pm0.23 (rm syst.))times10^{-6}$. The $90%$ confidence level upper limits on the branching fractions of the other modes having a significance of less than $3sigma$ are determined. These results are consistent with theoretical calculations using the nonrelativistic QCD factorization approach.