  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.
