We study the one-dimensional quarter-filled extended Hubbard model with an alternating transfer integral. In the strong-dimerization limit the charge part is described by the quantum Ising model which shows the two-dimensional Ising criticality at the self-dual point, and it is naturally connected to the double-frequency sine-Gordon theory in the weak dimerization. Treating low-lying excitations in finite-size systems, we numerically determine a phase boundary between two types of 4k(F) density-wave states and clarify the ground-state phase diagram. Further, we refer to its relevances to the charge-ordered phase observed in the charge-transfer organic salts.