Let $\mathbb A$ and $\mathbb B$ be relational structures over the same signature. A function $f\colon A\to B$ preserves a $k$-ary relation $R$ if for every $(a_1,\dots,a_k)\in R^{\mathbb A}$ we have that $(f(a_1),\dots,f(a_k))\in R^{\mathbb B}$.
Is there a commonly used notation for $f$ preserves $R$? Something like $f\models R$, $f\vdash R$, or $f \operatorname{pre} R$.
On
If $\Delta$ is a set of formulas, we sometimes write $f\colon A\to_\Delta B$ to indicate that all formulas in $\Delta$ are preserved by the map $f$ (see, for example, Section 3.1 of A Course in Model Theory by Tent and Ziegler).
So you could write $f\colon A\to_{\{R\}} B$ or $f\colon A\to_R B$ to denote that $f$ preserves $R$. But I've never seen this notation used in the case of a single relation symbol - it's much more common to just write "$f$ preserves $R$" in words. As Rick points out in the other answer, homomorphisms and embeddings (by definition) preserve all of the relation symbols in the language, so it's unusual to consider a map between structures which doesn't preserve all relation symbols.
I strongly recommend against the notations $f\models R$ and $f\vdash R$. The symbols $\models$ and $\vdash$ are already overloaded in model theory, and the idea of preservation is conceptually quite different from the usual meanings of these symbols (satisfaction, entailment, and provability).
A map between relational structures over the same signature is exaclty the same as a homomorphism between these structures, so I would just say that $f$ is a homomorphism; I'm not aware of any other notation for that.