It doesn't quite make sense to speak of "birational equivalence" of arbitrary Riemann surfaces. For instance, is the upper half plane "birationally equivalent" to $\mathbb{P}^1$?
Even if you restrict to nonsingular, connected algebraic curves over $\mathbb{C}$, the answer is no. Consider for instance the affine curve $C$ obtained as $\mathbb{P}^1$ with the points $0,1,\infty,\pi$ removed. The moduli space of curves of genus zero with four points removed is isomorphic to the affine line -- one can view this in terms of the cross ratio, or in terms of building an elliptic curve ramified over precisely those four points on $\mathbb{P}^1$. As soon as you have a positive dimensional moduli space, you have "generic points" which cannot be defined over $\overline{\mathbb{Q}}$. The curve $C$ has transcendental cross-ratio / $j$-invariant, so is such an example.