[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 (a surface of general type with the same rational cohomology as $\mathbb{P}^1 \times \mathbb{P}^1$) with universal cover the product of two hyperbolic planes will also have the desired property. You should also look into other surfaces with $p_g = q = 0$.


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