tldr: For a UV/Vis spectra, it shouldn't matter much.
I've found that Gaussian's AM1 / PM3 implementation often breaks symmetry during optimization. So I'm not surprised at your finding.
Moreover, in many cases, it's good to start with the expected symmetry geometry. For example, if I open the output in Avogadro, I can use Properties => Symmetry => Symmetrize to correct to $D_{2h}$ .. it seems very, very close. GaussView has a similar feature, as do many other interfaces (WebMO, etc.)
As mentioned in the comment, optimizations can break symmetry. In Gaussian, you can use, e.g.:
Symm=(Loose,Follow)
This loosens the symmetry perception in the initial step and requests that the point group (hopefully $D_{2h}$) is retained during optimization.
When you convert to a z-matrix, you can also force a symmetry by using symbolic notation to ensure corresponding bonds and angles are kept equivalent. The documentation gives the example of hydrogen peroxide:
H
O 1 R1
O 2 R2 1 A
H 3 R1 2 A 1 D
Variables:
R1 0.9
R2 1.4
A 105.0
D 120.0
Notice that the two H-O-O angles are defined as the same variable (A), and the H-O bond lengths are defined as identical (R1), ensuring symmetry no matter the optimized values.
It's a bit tedious, but you could do this for your benzoquinone. It looks like some of the bond angles are a tiny bit off, which is why Gaussian has optimized it into the $C_s$ point group.
Your question, though, is whether it will matter for UV/Vis spectra. In my experience, due to orbital symmetry, you might see some issues with excitation intensities (i.e., transitions that might be "forbidden" in the higher symmetry, but weak in the low-symmetry group). But in real life, molecules are dynamic, leading to such symmetry breaking anyway. So a symmetry-forbidden UV/Vis transition is usually just a weak excitation in an experimental spectra.
I'd worry much more about getting an accurate geometry and potential errors in your choice of methods. I don't remember G09 that well, but you should try to use a larger basis set (e.g., def2-TZVP) and ideally a dispersion-corrected functional (B3LYP-D3BJ if you want B3LYP) for the geometry. I'd also look at different methods for your TD-DFT calculations. This will matter more than a geometry in a symmetric subgroup.