We calculate spin and charge excitation spectra of the one-dimensional (1D) quarter-filled Hubbard model with nearest-neighbor
t and next-nearest-neighbor ${t}^{\ensuremath{'}}$ hopping integrals using the dynamical density-matrix renormalization group technique. We consider a case where
t (>0) is much smaller than ${t}^{\ensuremath{'}}$
(>0). First, we examine the accuracy of our method based on comparison between our result and exact noninteracting spectrum. Next, we investigate the spectra with the on-site Coulomb interaction. We find that the spin and charge excitation spectra are essentially the same as those of the 1D quarter-filled Hubbard (and $t\text{\ensuremath{-}}J$) model for the two 1D chains along the hopping integral ${t}^{\ensuremath{'}}$. However, the hopping integral
t $(<{t}^{\ensuremath{'}})$ plays a crucial role in the short-range correlations and low-energy excitaions; ferromagnetic correlation between electrons on neighboring sites is enhanced and pairing correlation between the electrons is induced. Consequently, a spin-triplet superconducting state may be derived.