We report 75As-NQR/NMR studies on the oxygen-deficient iron(Fe)-based oxypnictide superconductors LaFeAsO_{0.6} (T_c=28 K) along with the results on LaFeAsO, LaFeAsO_{0.75}(T_c=20 K) and NdFeAsO_{0.6}(T_c=53 K). Nuclear spin-lattice relaxation rate 1/T_1 of 75As NQR at zero field on LaFeAsO_{0.6} has revealed a T^3 dependence below T_c upon cooling without the coherence peak just below T_c, evidencing the unconventional superconducting state with the line-node gap. We have found an intimate relationship between the nuclear quadrupole frequencyof 75As and T_c for four samples used in this study. It implies microscopically that the local configuration of Fe and As atoms is significantly related to the T_c of the Fe-oxypnictide superconductors, namely, the T_c can be enhanced up to 50 K when the local configuration of Fe and As atoms is optimal, in which the band structure may be also optimized through the variation of hybridization between As 4p orbitals and Fe 3d orbitals.