Let $\newcommand{\Nplus}{\mathbb{N}^+}\Nplus$ denote the set of positive integers. Is there a function $f:\Nplus\to\Nplus$ with the following property?
For all $(a,b)\in \Nplus\times\Nplus$ there is a unique $(x,y) \in \Nplus\times\Nplus$ with $x < y$ and $(a,b) = (|x-y|,|f(x) - f(y)|)$?
Yes. Denote $X_k=(k, f(k))\in \newcommand{\Nplus} {\mathbb{N}^+}\Nplus\times \Nplus$. We want to construct the point set $A:=\{X_1,X_2,\ldots\}\subset \Nplus\times \Nplus $ so that:
$A$ contains exactly one point on each vertical line $\{i\}\times \Nplus$;
$A$ is a Sidon set: all differences between elements of $A$ are distinct;
every vector $(a, b) \in \Nplus\times \Nplus$ is realized as a difference of two elements of $A$.
This may be done by a standard procedure with adding points to $A$. On $(2k-1)$-th step you add a point on the $k$-th vertical line if it still does not exist; on $2k$-th step you add two points whose difference is the $k$-th vector from the set $\Nplus\times \Nplus$ (enumerated in advance) if there are still no two such points; and always care on property 2.
