I think following post in the mathoverflow gives an answer:

https://mathoverflow.net/questions/79046/effective-versus-movable-cones-of-curves

There, people mention that there is an example where the Ample cone is rational polyhedral but movable cone is not. But if Ample cone is polyhedral then $\overline{Q}(X)$ would be polyhedral too and so they can not be equal.