Yes. Use the same construction as the Rado graph, but replace the base 2 by base 4. The vertices are in one-to-one correspondence with the natural numbers, with edges obeying the following rule.
For any two vertices $x,y$ with $x:
If the $x$th bit of the base-4 expansion of $y$ is a $0$, then there are no edges between $x$ and $y$.
If the $x$th bit of the base-4 expansion of $y$ is a $1$, then there is a directed edge from $x$ to $y$.
If the $x$th bit of the base-4 expansion of $y$ is a $2$, then there is a directed edge from $y$ to $x$.
If the $x$th bit of the base-4 expansion of $y$ is a $3$, then there are directed edges in both directions between $x$ and $y$.
This has very similar properties to the Rado graph, and roughly the same algorithm works for finding induced subgraphs of a given isomorphism type.
Incidentally, the idea here is presumably the same as the construction of 4-edge-colored Rado graph $G_4$ mentioned in "related concepts" section of the same Wikipedia article.