Let $G$ be the free group with two generators, say $a$ and $b$. Then the Cayley of $G$ with respect with the generating set is 4 regular tree. If we contract every edge with the label $b$, then we end up with a tree $T$ such that $G$ acts on the edges of $T$ transitively.
Applying Bass-Serre theory implies that $G$ is the free product $\langle a\rangle $ and $\langle b\rangle $.
My question is the following:
How can I modify the Cayley graph of $G$ in order to get an HNN-exentension?
Actually, the tree that you have described is already an HNN extension tree, because its quotient graph of groups is a graph with one vertex and one edge. The vertex is labelled with the cyclic group $\langle b \rangle$, and the edge is labelled with the trivial group $E$.
The group element $a$ is the stable letter of that quotient edge, leading to the HNN presentation $$F_2 = \langle a,b \mid aea^{-1}=e \rangle $$ where $e$ is the identity element of $\langle b \rangle$.
Although there are other seemingly less trivial HNN extensions of $F_2$ that can be described, and although this may seem like a somewhat trivial example of an HNN extension, it is nonetheless an example.