Derived Category of certain maximal order on $\mathbb{P}^{2}$

We show that the moduli space of $A$-line bundles with minimal second Chern class is a fine moduli space, where $A$ is a maximal quaternion order on $\mathbb{P}^{2}$ ramified along a smooth quartic. We prove that there is a fully faithful embedding from the derived category of this moduli space into the derived category of $A$-modules. Furthermore, we find a semiorthogonal decomposition for $D^{b}(\mathbb{P}^{2},A)$.