Theorem 1.2 in Q. Xie `[On Pseudo-Effectivity of the Second Chern Classes for Terminal Threefolds][1]' shows that $c_2(X)$ is at least *pseudoeffective* (i.e., a limit of effective classes). On the other hand, since $X$ is Fano, the Cone theorem shows that the cone of curves is rational polyhedral, spanned by finitely many classes of rational curves, so $c_2(X)$ is in fact effective.  


  [1]: https://arxiv.org/pdf/math/0409269.pdf