Mathematical symbol to show one mathematical structure represents another

The short answer is no: there is no standard symbol for this.

The long answer is that you could use a symbol (such as $M_R \sim R$ as suggested by Torsten in the comments), or introduce some other kind of notation such as $[R] = M_R$, but if you wanted to do this then it would be a good idea to specify explicitly what the notation means. For example:

In what follows, we will write $M \sim R$ to denote the assertion that the relation $R$ is represented by the connection matrix $M$.

There is no universal such symbol as far as I know - and in my opinion, there shouldn't be: there are many different ways one object can "represent" another, and we should distinguish between them. I think being overly concise can be a negative here.

Remember also that you can always introduce new terminology/notation in a paper, e.g. "When we have $aRb$ iff the $\langle a,b\rangle$-entry of $M$ is $1$, we write "$R\sim M$.""