This paper is dedicated to the study of ${\mathcal{H}}_{\mathrm{\infty }}$ synchronization for Hopfield networks (HNs) with time-variable delay via quantized control. The delay function is assumed to be continuous and bounded, yet not necessarily differentiable. A delay-dependent sufficient condition is derived using a linear combination of Lyapunov functional candidates along with two matrix inequalities to ensure the ${\mathcal{H}}_{\mathrm{\infty }}$ synchronization of drive-response HNs. Based on this, an alternative sufficient condition characterized by reduced nonlinearities is further established by employing decoupling techniques. With fixed bounds on the time delay and the ${\mathcal{H}}_{\mathrm{\infty }}$ disturbance attenuation level, an algorithm is then proposed to compute the minimum required gain for the quantized controller. Finally, a numerical example demonstrates the effectiveness of the proposed ${\mathcal{H}}_{\mathrm{\infty }}$ synchronization results.