We report the first search for the $J^{PC}=0^{--}$ glueball in $Upsilon(1S)$ and $Upsilon(2S)$ decays with data samples of $(102pm2)$ million and $(158pm4)$ million events, respectively, collected with the Belle detector. No significant signals are observed in any of the proposed production modes, and the 90% credibility level upper limits on their branching fractions in $Upsilon(1S)$ and $Upsilon(2S)$ decays are obtained. The inclusive branching fractions of the $Upsilon(1S)$ and $Upsilon(2S)$ decays into final states with a $chi_{c1}$ are measured to be $BR(Upsilon(1S)to chi_{c1}+ anything) = (1.90pm 0.43(stat.)pm 0.14(syst.))times 10^{-4}$ with an improved precision over prior measurements and $BR(Upsilon(2S)to chi_{c1}+ anything) = (2.24pm 0.44(stat.)pm 0.20(syst.))times 10^{-4}$ for the first time.