The quivers of surface type $D$ are constructed similarly to type $A$ but instead take place within a regular $n$-gon with one of the triangulation vertices sitting in the interior of the $n$-gon. See Figure 5.8 of Introduction to Cluster Algebras. Chapters 4-5. We can also add the arrows between the frozen vertices of the quiver without changing the cluster algebra to get a balanced quiver. The resultant quiver will be restricted mutation-finite and the cluster variables generated through sequences of restricted mutations give rise to the type $D$ cluster algebras (with coefficients).