Answer Set Programming with Quantifiers ASP(Q) is a recent extension of Answer Set Programming (ASP) that allows one to model problems from the entire polynomial hierarchy. Earlier work focused on demonstrating modeling capabilities of ASP(Q). However, further evidence of modeling efficacy of \QASP is needed and, even more importantly, effective solvers for ASP(Q) must be developed. In this paper, we address both these areas. First, we provide elegant quantified ASP encodings for additional relevant AI problems, such as \emph{Argumentation Coherence} ($\Pi_2^P$-complete) and \emph{Semi-stable Semantics for ASP} ($\Sigma_2^P$-complete). Second, we provide a modular ASP(Q) solver that translates a quantified ASP program together with a given data instance into a QBF to be solved by any QBF solver. We evaluate the performance of our solver on several instances and with different back-end QBF solvers, demonstrating its efficacy as a tool for rapid modeling and solving of complex combinatorial problems.