I take it morphisms $f: X \to Y$ are by definition functions that preserve the relation: if $x, x'$ are related in $X$, then $f(x), f(x')$ are related in $Y$.

It's easy to manufacture some silly examples of nontrivial monads, by finding suitable monoidal products on the category of graphs and then finding monoids with respect to that monoidal product. One such monoidal product takes the disjoint sum of two graphs. Then an example of a monoid therein is the one-point graph $1$, which carries a unique monoid structure.

The associated monad $M$ takes a graph and adjoins an isolated point to the graph, which one could regard as basepoint. If $f: X \to Y$ is a graph morphism, then $M(f)$ is the obvious basepoint preserving extension. To be precise, $M(X) = 1 + X$; the unit of the monad is the inclusion of $X$ in $1 + X$, and the multiplication $1 + 1 + X \to 1 + X$ is the identity on $X$ but identifies the two copies of $1$ as one.