This is a variation on Poonen's question, taking Buzzard's fabulous example into account :

Question 1. Is there a smooth proper scheme $X\to\operatorname{Spec}(\mathbb{Z})$ such that $X(\mathbb{Q}_v)\neq\emptyset$ for every place $v$ of $\mathbb{Q}$ (including $v=\infty)$, and yet $X(\mathbb{Z})=\emptyset$ ?

Question 2. Is there such an example over (the ring of integers of) some number field ?

Remark 1. Let $K$ be a real quadratic field, $\mathfrak{o}$ the ring of integers of $K$, and $A$ the quaternion algebra over $K$ which is ramified exactly at the two real places. Then the conic $C$ corresponding to $A$ is a smooth proper $\mathfrak{o}$-scheme such that $C(\mathfrak{o})=\emptyset$.

Remark 2. Colliot-Thélène and Xu give examples of affine $\mathbb{Z}$-schemes which have points everywhere locally, but no $\mathbb{Z}$-points. Some of these schemes might even be smooth.