Periodic nanoparticle arrays with lower loss or high-quality (Q) factor resonances have made great advances in numerous applications. Such arrays can support narrow resonance modes by the hybridization of plasmonic-photonic modes, known as lattice plasmon modes (LPMs). The LPMs arise from the diffraction-induced coupling between localized surface plasmon resonances (LSPRs) of nanoparticles in symmetric or quasi-symmetric refractive index environment. To date, the great majority of researches have focused on the coupling of dipolar modes to in-plane orthogonal diffraction waves in nanoparticle arrays, resulting in dipolar lattice plasmon modes (DLPMs). However, there are few studies about quadrupolar lattice plasmon modes (QLPMs) for parallel coupling in symmetric environment, especially for high Q-factor QLPMs. In this work, we explore the sharp QLPMs of the silver nanodisk arrays by x-polarized light at normal incidence. In the first place, the scattering cross-section and near-field electric field distribution of single silver nanodisk indicate the existence of dipolar and quadrupolar LSPRs, thus, the optical responses of silver nanodisk arrays exhibit the peak-and-dip profiles of DLPMs and QLPMs at different wavelengths. Also, the electromagnetic field distributions confirm that the parallel propagating electric field introduces QLPM and results in electric field delocalization, while DLPM is existent in another way in periodic silver nanodisk arrays. Moreover, the position, linewidth and lineshape of the QLPM strongly depend on the role of lattice period. We enable these resonance modes to be selectively accessed and individually optimized by tuning lattice periods in the x- or y-direction. By changing the lattice period in the x-direction from 300 to 550 nm in steps of 50 nm, the dip of transmission intensity increases gradually, and when periods in the two directions are equal, the transmission dip exhibits a narrow-band QLPM resonance with a linewidth of 0.4 nm, corresponding quality factor is as high as Q = 1815 under the x-polarized light. In particular, by varying periods in the y-direction, the QLPM can also be manipulated in a range from an asymmetric Fano-like lineshape peak to a dip. The acquisition of these results may provide a design strategy for high-Q factor resonance in nanolaser, sensing, and nonlinear optics.