Reposting an answer from a deleted account:
Yes, there is an analogous Dehn-like presentation for knot quandles where generators correspond to regions of a knot diagram instead of arcs. This structure is called a "biquasile" and was introduced by Needell and Nelson in their paper “Biquasiles and Dual Graph Diagrams".
A biquasile is defined on a set $X$ with two binary operations $*$ and $\cdot$, each defining a quasigroup structure on $X$. The key idea is to associate biquasile elements with regions of a knot diagram, rather than arcs.
The biquasile axioms are motivated by the Reidemeister moves on dual graph diagrams. For all $a,b,x,y \in X$, we have:
$$a * (x \cdot [y * (a \cdot b)]) = (a * [x \cdot y]) * (x \cdot [y * ([a * (x \cdot y)] \cdot b)])$$ $$y * ([a * (x \cdot y)] \cdot b) = (y * [a \cdot b]) * ([a * (x \cdot [y * (a \cdot b)])] \cdot b)$$
These axioms ensure that biquasile colorings of dual graph diagrams are preserved under Reidemeister moves.
The fundamental biquasile of a knot or link $L$ is defined using a presentation $\langle X | R \rangle$, where $X$ is the set of regions in a diagram of $L$, and $R$ contains relations of the form:
$$\begin{cases} z = x * (b \cdot a) \\ x = z * (b \cdot a) \end{cases}$$
for each crossing in the dual graph diagram, where $x$, $z$ are opposite regions at the crossing, and $a$, $b$ are the other two adjacent regions.
An important example is the Alexander biquasile, defined on an $L$-module $X$ where $L = \mathbb{Z}[d^{\pm1}, n^{\pm1}, s^{\pm1}]$, with operations:
$$x * y = (-dsn^2)x + ny \quad \text{and} \quad x \cdot y = dx + sy$$
This structure generalizes the Dehn presentation of the knot group in a way analogous to how quandles generalize the Wirtinger presentation.