For each \(0<\alpha <\frac{1}{2}\) , there exists a Bayer–Lahoz–Macrì–Stellari inducing Bridgeland stability condition \(\sigma (\alpha )\) on a Kuznetsov component \(\textrm{Ku}(Q)\) of the smooth quadric threefold Q. We obtain the non-emptiness of the moduli space \(M_{\sigma (\alpha )}([{\mathcal {P}}_{x}])\) of \(\sigma (\alpha )\) -semistable objects in \(\textrm{Ku}(Q)\) with the numerical class \([{\mathcal {P}}_{x}]\) , where \({\mathcal {P}}_{x}\in \textrm{Ku}(Q)\) is the projection sheaf of the skyscraper sheaf at a closed point \(x\in Q\) . We show that the moduli space \({\overline{M}}_{Q}({\textbf{v}})\) of Gieseker semistable sheaves with Chern character \({\textbf{v}}=\textrm{ch}({\mathcal {P}}_{x})\) is smooth and irreducible of dimension four, and prove that the moduli space \(M_{\sigma (\alpha )}([{\mathcal {P}}_{x}])\) is isomorphic to \({\overline{M}}_{Q}({\textbf{v}})\) . As an application, we show that the quadric threefold Q can be reinterpreted as a Brill–Noether locus in the Bridgeland moduli space \(M_{\sigma (\alpha )}([{\mathcal {P}}_{x}])\) . In the appendices, we show that the moduli space \(M_{\sigma (\alpha )}([S])\) contains only one single point corresponding to the spinor bundle S and give a Bridgeland moduli interpretation for the Hilbert scheme of lines in Q.
