In this paper we continue the study of the spaces $h(p,q,\varphi)$ and $H(p,q,\varphi)$. We apply the main results of Part I to obtain new information on the coefficient multipliers of these spaces. For example, we find the multipliers from $h(p,q,\varphi)$ to $h(\infty,q_0,\varphi)$ for any $p\geq 1$, $q,p_0>0$ and any quasi-normal function $\varphi$, and this improves and generalizes a result of Shields and Williams [16]. We also describe the multipliers from $H(p,q,\alpha)$, $p\leq 1$, to $H(p_0,q_0,\alpha)$, $p_0\geq p$, and $l^s,\,s> 0$.