Let $X$ and $X'$ be two birational algebraic curves over $\mathbb{Q}$.
Is it true (or not) that $X(\mathbb{Q})$ is Zariski dense iif $X'(\mathbb{Q})$ is Zariski dense ?
If no, where can I find a counter-example?
If yes, what about the higher dimensional varieties?
This is true for any two birational varieties, assuming the birational map is defined over $\Bbb Q$. Suppose $X(\Bbb Q)\subset X$ is Zariski dense. We will show that any nonempty open subset $V'\subset X'$ contains a rational point.
Recall that two varieties $X,X'$ are birational iff there are dense open sets $U\subset X$ and $U'\subset X'$ with $U\cong U'$. Fix $U,U'$ as in the previous sentence; then $V'\cap U'$ is a nonempty open subset as $X'$ is irreducible, so $V'\cap U'$ is $\Bbb Q$-isomorphic to a nonempty open subset of $X$ and therefore contains a rational point by density. Since a $\Bbb Q$-isomorphism preserves rational points, we see that $V'\cap U'$ contains a rational point, hence $V'$ contains a rational point, and we win.