Remote sensing of atmospheric refractive index structure constant ($\boldsymbol{C}_{\boldsymbol{n}}^2$) using lidar incorporating a single-photon detector (SPD) is proposed. The influence of turbulence on the fiber coupling efficiency with different fiber modes is analyzed. $\boldsymbol{C}_{\boldsymbol{n}}^2$ can be derived from the ratio of the backscattering signals counted on single-mode and multimode fiber-coupling channels of the SPD. In the experiment, by eliminating the shot noise effect on the fluctuation of the ratio, the lowest coupling ratio is used to retrieve $\boldsymbol{C}_{\boldsymbol{n}}^2$ and demonstrated by comparing to the results measured from a large aperture scintillometer (LAS). Good agreement between results from the LAS and the lidar is achieved. The correlation coefficients are 0.90, 0.89, and 0.89, under three different weather conditions.