  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 $m_\pi = 142$
MeV. By adopting the infinite-volume reconstruction method along with an
optimized subtraction scheme, we obtain $\Delta E_{\text{TPE}} =
-28.9(4.9)~\mu\text{eV} + 93.72~\mu\text{eV}/\text{fm}^2 \cdot\langle r_p^2
\rangle$, or $\Delta E_{\text{TPE}} = 37.4(4.9)~\mu$eV, which is consistent
with the previous theoretical results in a range of 20-50 $\mu$eV.
