The title of your question asks about "exponential object[s] in the category of simple, undirected graphs". I can tell you what they are.

(The body of your question asks about a construction in a specific paper, which I haven't read, so I can't answer that question directly. But of course, exponentials are unique when they exist.)

So: in the category of simple graphs that you mention, the exponentials $\mathrm{HOM}(G_1, G_2)$ are as follows. A vertex of $\mathrm{HOM}(G_1, G_2)$ is a graph homomorphism from $G_1$ to $G_2$. Two homomorphisms $\phi, \psi: G_1 \to G_2$ are adjacent in $\mathrm{HOM}(G_1, G_2)$ iff whenever $x$ and $y$ are adjacent vertices of $G_1$, then $\phi(x)$ and $\psi(y)$ are adjacent vertices of $G_2$.

I'd recommend Godsil and Royle's book Algebraic Graph Theory. This blog post also says more about the cartesian closed category of simple graphs.

The title of your question asks about "exponential object[s] in the category of simple, undirected graphs". I can tell you what they are.

(The body of your question asks about a construction in a specific paper, which I haven't read, so I can't answer that question directly. But of course, exponentials are unique when they exist.)

So: in the category of simple graphs that you mention, the exponentials $\mathrm{HOM}(G_1, G_2)$ are as follows. A vertex of $\mathrm{HOM}(G_1, G_2)$ is a function from the set of vertices of $G_1$ to the set of vertices of $G_2$. Two vertices $\phi, \psi$ of $\mathrm{HOM}(G_1, G_2)$ are adjacent iff whenever $x$ and $y$ are adjacent vertices of $G_1$, then $\phi(x)$ and $\psi(y)$ are adjacent vertices of $G_2$.

I'd recommend Godsil and Royle's book Algebraic Graph Theory. This blog post also says more about the cartesian closed category of simple graphs.