The infinite time-evolving block decimation algorithm is applied to calculate the dynamical spinstructure factors of the quasi-one-dimensional (Q1D) S=1 antiferromagnetic spin system with singleion anisotropy and bond alternation. It is found that when the staggered field induced by the weak interchain interaction is taken into account, the excitation continuum originating from magnons is quantized. The excitation energies of the quantized excitation spectra are well-explained by the negative zeros of the Airy functions, when the single-ion anisotropy is negatively strong. This quantization of the magnon continuum is a counterpart of the spinon confinement, which has been recently discussed for Q1D S=1/2 antiferromagnets. It is further shown that, when a staggered field exists, the quantized excitation spectra appear on the phase boundary between the Haldane and Néel phases of the phase diagram without the staggered field. However, the quantized excitation spectra disappear in the singlet-dimer phase. PACS numbers: 75.10Pq, 75.40Gb, 75.40Mg Recent inelastic neutron scattering (INS) experiments 1,2on quasi-one-dimensional (Q1D) S=1/2 antiferromagnetic (AF) Ising-like XXZ magnets, (Ba/Sr)Co 2 V 2 O 8 3 , have reported that, below the Néel temperature T N , these materials show quantized excitation spectra with excitation energies that are well explained by the series of negative zeros in the Airy function (NZAF). McCoy and Wu argued the relation between the NZAF and confinement of the domain-wall excitations by focusing on the excitation spectra of the S=1/2 ferromagnetic (FM) Ising chain in weak transverse and longitudinal magnetic fields 5,13 . Since the ground state of this model is a fully polarized state, low-energy excitation is achieved by flipping spins of arbitrary lengths, i.e., two domain-wall excitations. Since each domain wall carries ∆S = 1/2, the low-energy excitation is interpreted as two-spinon excitation. In the transverse field, these two spinons travel in the chain and compose an excitation continuum. When the uniform longitudinal field is further applied weakly, it works as a linear potential between the two spinons. The effective model for this two-spinon excitation is exactly solvable and the eigenvalues are given by NZAF 5,13 . This means that the excitation spectra of the two-spinon continuum are quantized. The quantized spectra explained by NZAF have been confirmed in the INS experiments 6 on CoNb 2 O 6 , which is a ferromagnetic Ising-spin-chain compound. Shiba also discussed the quantized excitation spectra in the Q1D S=1/2 AF Ising-like XXZ system 14 . When we focus on bipartite systems, the interchain interaction in the Q1D AF systems effectively works as a weak staggered field below T N . Thus, it is expected that the domain-wall excitations along the spin-chain direction are confined by the weak staggered field and that the same discussion as that for the ferromagnetic Ising-spin chain is applicable for low-energy excitation.The two-spinon confinement plays a key role in the quantize...