In this paper, we propose a new approach to probe the quarkonium at the BES-III and the B factories. In the two-photon process, e(+)e(-) -> e(+)e(-) + hadron, the signals of the emitted hadron can be reconstructed through investigating the final leptons e(+)e(-). Furthermore, by analyzing the distribution of the relative angle between the final e(+) and e(-), we can determine the quantum number of the final hadron. For those hadrons that are hard to detect directly, this may provide an effective way. We calculate the cross sections as well as the distributions of the eta(c)(1S, 2S) and chi(cJ)(1P, 2P) production at tree level and the next-to-leading order QCD radiative corrections. In addition, the cross sections of bottomonium production are considered at tree level. Unfortunately, they are too tiny to be detected.