[Fake projective planes][1] have $H_2(X,\mathbb{Z}) \cong \mathbb{Z}$. They have metrics of pinched negative curvature, so they have $\pi_2(X) \cong \{0\}$. Thus $H_2(X,\mathbb{Q}) \cong \mathbb{Q}$ is generated by a  hyperplane section, and this is not a rational curve.

A Picard maximal fake quadric with universal cover the product of two hyperbolic planes will also have the desired property.


  [1]: https://en.wikipedia.org/wiki/Fake_projective_plane