  Given a quantum graph $\mathcal{G}=(B,\psi,A)$, we define a C*-correspondence
$E_\mathcal{G}$ over the noncommutative vertex C*-algebra $B$, called the
quantum edge correspondence. For a classical graph $\mathcal{G}$,
$E_\mathcal{G}$ is the usual graph correspondence spanned by the edges of
$\mathcal{G}$. When the quantum adjacency matrix $A\colon B\to B$ is completely
positive, we show that $E_\mathcal{G}$ is faithful if and only if $\ker(A)$
does not contain a central summand of $B$. In this case, we show that the
Cuntz-Pimsner algebra $\mathcal{O}_{E_\mathcal{G}}$ is isomorphic to a quotient
of the quantum Cuntz-Krieger algebra $\mathcal{O}(\mathcal{G})$ defined by
Brannan, Eifler, Voigt, and Weber. Moreover, the kernel of the quotient map is
shown to be generated by "localized" versions of the quantum Cuntz-Krieger
relations, and $\mathcal{O}_{E_\mathcal{G}}$ is shown to be the universal
object associated to these local relations. We study in detail some concrete
examples and make connections with the theory of Exel crossed products.
