Fake projective planes 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.