We report a characterization of an X-ray-detected quiescent galaxy at $z=2.09$, named COS-XQG1, using JWST/NIRCam and NIRSpec data. This galaxy is detected in Chandra imaging, suggesting the presence of an AGN with a high black hole accretion rate of ṀBH=0.22±0.03\, M_ yr⁻¹. Using multi-wavelength photometry from X-ray to sub-millimeter, including the latest JWST imaging, we confirm that COS-XQG1 is massive (M_ = (1.6±0.2)×10¹¹\, M_) and quiescent (sSFR=(0.9± 1.8)×10⁻¹¹\, yr⁻¹) as reported previously, even considering the contribution from AGN emission. Noticeably, COS-XQG1 displays a broad line Hα emission component with a full width at half maximum of 4491⁺¹¹⁸₋₁₁₀\, km\, s⁻¹ in its NIRSpec spectrum. The line width and luminosity of the broad Hα emission give a black hole mass of log(MBH/M₎ = 8.45±0.02\, (± 0.5). With a stellar velocity dispersion measurement (σ₌235±35\, km\, s⁻¹), we find that this galaxy is consistent with the local relations in the MBH - σ_ and MBH- M_ planes, which might suggest that massive quiescent galaxies at z≥2 have already been mature in terms of both stellar and black hole masses and will not evolve significantly. In addition, image 2D-decomposition analysis finds that this galaxy comprises disk and point source components. The latter is likely the composition of an AGN and a stellar bulge. Based on a comparison with numerical simulations, we expect that COS-XQG1 will evolve into a typical bulge-dominated quiescent galaxy with lower AGN activity by redshift 0. This study shows the usefulness of X-ray-detected quiescent galaxies in investigating the co-evolution between SMBHs and galaxies in the early Universe.
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