This sounds wrong to me. I think $D^b(Q)$ should be replaced by the derived category of finite length modules over the corresponding preprojective algebra of affine type.
Homological mirror symmetry would say the derived Fukaya category should be equivalent to the bounded derived category of quasi-coherent sheaves over the mirror manifold (in this case, over the exceptional fibre of the resolution of the singularity). I don't understand this in any detail, and I don't know a reference for it, so I may not be getting this quite right.
The quasi-coherent sheaves supported on the exceptional fibre form an abelian category equivalent to the finite length modules of the preprojective algebra of the corresponding affine type. This is discussed in the introduction Bridgeland's Stability conditions and Kleinian singularities. The reference he gives for the link between the two is Crawley-Boevey and Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), no. 3, 605–635.