Ellipticity properties of high-order harmonic generation (HHG) from symmetric molecules $ {\text{H}}_{\text{2}}^ + $ in strong and short wavelength (less than 800nm) laser fields are numerically investigated. In this study, the ellipticity of harmonic is compared with the corresponding harmonic spectrum and dipole, and the calculation results are analyzed and the results obtained at different laser intensities, different laser wavelengths, different internuclear distances and different orientation angles are compared with each other. Our numerical simulations show that the influences of laser intensity, laser wavelength, internuclear distance and orientation angle on the ellipticity of harmonic are different. Especially in a two-center interference region, the excited state plays an important role in the HHG, but the effects of the excited state on the ellipticity of harmonic are different at different orientation angles. Further analysis shows that these different effects are due to the influence of the excited state on the harmonic yield. Using the numerical scheme, it is determined that in the two-center interference region, the excited state plays an important role in the parallel harmonic spectrum, while the effects of the excited state on the perpendicular harmonics at different angles are all very small, which results in different phase differences between the accurate harmonic spectrum and the harmonic spectrum only returning to the ground state. Overall, the relative yields of the accurate perpendicular harmonics are lower (higher) than those of the accurate parallel harmonics, but the intensities of the perpendicular harmonics, which only return to the ground state, are comparable to (or farther away from) those of the parallel harmonics which are only to return to ground state in the two-center interference regions. Therefore, the small (large) intensity ratio between the accurate perpendicular harmonic and accurate parallel harmonic can be attributed to the contributions of the excited state to harmonics. Then we can conclude that the harmonic spectra that only go back to the ground state show high (small) ellipticity, whereas the accurate harmonic spectra show small (high) ellipticity, resulting in a strong angle dependence of the influence of the excited state on the ellipticity of harmonic. In addition, in the high-order harmonic plateau region, the relative yields of harmonics can be well predicted by the corresponding dipoles, indicating the applicability of tunneling pictures and plane wave approximation in the strong and short-wave laser fields. When the ellipticity of harmonic occurs in the interference region due to the two-center characteristics of the symmetric potential, the results show that the polarization measurement can also be used to detect the structures of symmetric molecules and track the dynamic behaviors of excited states.