= Solution
Each factor is a <free abelian group> of rank two. Take the infinite symmetric generating set
$$
\boxed{S=(A\setminus\{1\})\cup(B\setminus\{1\}).}
$$
The <Bass-Serre tree> has an edge $e_g$ joining $gA$ to $gB$ for every $g\in A*B$. Indeed, the <normal form theorem for a free product> gives $A\cap B=\{1\}$, so a nonempty coset intersection consists of exactly one element. Let $m_g$ be the midpoint of $e_g$ and send the Cayley vertex $g$ to $m_g$.
Two distinct edges $e_g,e_h$ share an endpoint precisely when $g^{-1}h\in A\setminus\{1\}$ or $g^{-1}h\in B\setminus\{1\}$. This is exactly adjacency in the <Cayley graph> for $S$. A path in that graph therefore gives a path between the midpoints of the same length in the tree. Conversely, the unique tree segment between two edge midpoints passes through a succession of adjacent edges and gives a Cayley path of that length. Hence, on vertices,
$$
\boxed{d_S(g,h)=d_{\mathcal T}(m_g,m_h).}
$$
All edge midpoints occur, and every tree point is within $1/2$ of one of them. Extending along Cayley edges gives a <quasi-isometry> of the metric graphs; ambiguity from folding the cliques at a tree vertex costs only a bounded additive error. This is the <free product with factor generating set> construction.
For finite generating sets the answer changes. Use $S_0=\{a,a',b,b'\}^{\pm1}$. The retraction $A*B\to A$ killing $B$ sends every generator to a generator of $A$ or to the identity. Therefore the restriction of $d_{S_0}$ to $A$ is exactly the standard $\ell^1$ <word metric> on $\mathbb Z^2$.
In this grid, take vertices $(0,0),(2N,0),(0,2N)$. Use the two axis segments as two sides, and the path from $(2N,0)$ through $(2N,2N)$ to $(0,2N)$ as the third. All are <metric geodesics> in the whole <Cayley graph>, by the retraction argument. The corner $(2N,2N)$ is at distance $2N$ from either of the other sides. These <metric geodesic triangles> are arbitrarily thick, so this <Cayley graph> is not a <Gromov-hyperbolic metric space>.
A tree is $0$-hyperbolic. <Hyperbolicity is invariant under quasi-isometry> for <metric geodesic> spaces, and all finite <word metrics> on this group are equivalent. Consequently \b[no <Cayley graph> for a finite generating set is quasi-isometric to any tree], including a locally infinite one.
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