We report the first search for the J(PC) = 0(--) glueball in Upsilon(1S) and Upsilon(2S) decays with data samples of (102 +/- 2) x 10(6) and (158 +/- 4) x 10(6) 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 chi(c1) are measured to be B(Upsilon(1S) -> chi(c1) + anything) = (1.90 +/- 0.43(stat) +/- 0.14(syst) x 10(-4) with an improved precision over prior measurements and B Upsilon(2S) -> chi(c1) + anything) = (2.24 +/- 0.44(stat) +/- 0.20(syst) x 10(-4) for the first time.