I am looking for an example of a surface $X \subset \mathbb{P}^3$ defined over $\mathbb{Q}$ with the following properties: 1) $X(\mathbb{Q}) \ne \emptyset$; and 2) we know all of the elements $X(\mathbb{Q})$ explicitly. Here explicitly means that we can write $X(\mathbb{Q})$ as the union of rational points on an explicit finite collection of curves on $X$, and possibly a finite collection of explicitly determined rational points on $X$.
Note that the assumption that $X(\mathbb{Q}) \ne \emptyset$ is critical; certainly there are many surfaces $X$ which provably have no rational points (perhaps due to having a local obstruction, say).
Here is a conjectured example: for $n \geq 5$ consider the surface $X_n$ defined by
$$\displaystyle X_n : x_0^n + x_1^n = x_2^n + x_3^n.$$
Then $X_n$ contains the rational lines defined by $(x_0, x_1) = (x_2, x_3)$ and $(x_0, x_1) = (x_3, x_2)$, and it is a folklore conjecture that $X_n$ does not contain any other rational points.