Does there exist an elliptic curve $E/\Bbb{Q}$ such that for all quadratic number field $K$, $\DeclareMathOperator{\rank}{\operatorname{rank}} \rank(E/K)=0$ ?
Let $K=\Bbb{Q}(\sqrt{D})$, $\rank(E/K)=\rank(E/\Bbb{Q})+\rank(E_D/\Bbb{Q})$.
So $E/\Bbb{Q}$ need to satisfy $\rank(E/\Bbb{Q})=0$ and $\rank(E_D/\Bbb{Q})=0$ for all $D: \text{square free integer}$.