  Let $\Gamma$ denote a finite, undirected, connected graph, with vertex set
$X$. Fix a vertex $x \in X$. Associated with $x$ is a certain subalgebra
$T=T(x)$ of ${\rm Mat}_X(\mathbb C)$, called the subconstituent algebra. The
algebra $T$ is semisimple. Hora and Obata introduced a certain subalgebra $Q
\subseteq T$, called the quantum adjacency algebra. The algebra $Q$ is
semisimple. In this paper we investigate how $Q$ and $T$ are related. In many
cases $Q=T$, but this is not true in general. To clarify this issue, we
introduce the notion of quasi-isomorphic irreducible $T$-modules. We show that
the following are equivalent: (i) $Q \neq T$; (ii) there exists a pair of
quasi-isomorphic irreducible $T$-modules that have different endpoints. To
illustrate this result we consider two examples. The first example concerns the
Hamming graphs. The second example concerns the bipartite dual polar graphs. We
show that for the first example $Q=T$, and for the second example $Q \neq T$.
