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 projective $\mathfrak{o}$-scheme such that $C(\mathfrak{o})=\emptyset$ (because $C(K_v)=\emptyset$ for each of the real places $v$).

Remark 2. Colliot-Thélène and Xu give a systemetic treatment of quasi-projective $\mathbb{Z}$-schemes $Y$ such that $Y(\mathbb{Z}_p)\neq\emptyset$ for every prime $p$ and $Y(\mathbb{R})\neq\emptyset$, but $Y(\mathbb{Z})=\emptyset$. Some of these schemes might even be smooth, but none of them is proper.