We develop a method for lattice QCD calculation of the two-photon exchange contribution to the muonic-hydrogen Lamb shift. To demonstrate its feasibility, we present the first lattice calculation with a gauge ensemble at m_{π}=142 MeV. By adopting the infinite-volume reconstruction method along with an optimized subtraction scheme, we obtain with statistical uncertainty ΔE_{TPE}=-28.9(4.9) μeV+93.72 μeV/fm^{2}·⟨r_{p}^{2}⟩, or ΔE_{TPE}=37.4(4.9) μeV, which is consistent with the previous theoretical results in a range of 20-50 μeV.