In this paper, we associate quantum vertex algebras to a certain family of associative algebras $\widetilde{\A}(g)$
which are essentially Ding-Iohara algebras. To do this, we introduce another closely related family
of associative algebras $\A(h)$. The associated quantum vertex algebras are based on the vacuum modules
for $\A(h)$, whereas $\phi$-coordinated modules for these quantum vertex algebras are associated to
$\widetilde{A}(g)$-modules. Furthermore, we classify their irreducible $\phi$-coordinated modules.