We present measurements of the first to fourth moments of the lepton mass squared q2 of B→Xcl ̄νl decays for l=e, μ and with Xc a hadronic system containing a charm quark. These results use a sample of electron-positron collisions at the Υ(4S) resonance corresponding to 62.8 fb−1 of integrated luminosity and collected by the Belle II 2 experiment in 2019 and 2020. To identify the Xc system and reconstruct q2, one of the B mesons from an Υ(4S)→B ̄B decay is fully reconstructed in a hadronic decay mode using a multivariate B tagging algorithm. We report raw and central moments for q2>1.5 GeV2/c4 up to q2>8.5 GeV2/c4, probing up to 77% of the accessible B→Xcl ̄νl phase space. This is the first measurement of moments in the experimentally challenging range of [1.5,2.5] GeV2/c4. The results can be used for a new determination of |Vcb| using inclusive B→Xcl ̄νl decays.