In order to clarify the variation of the microscopic magnetic nature with the carrier density in Na_xCoO₂⋅ yH₂O, we have measured positive muon-spin rotation/relaxation (µ⁺SR) spectra for H₂O as well as D₂O absorbed samples. Based on the zero field (ZF-) µ⁺SR measurements, there was no clear difference between the two superconducting phases (SC-I and SC-II). Furthermore, the ZF-spectrum for the H₂O absorbed sample exhibits a clear oscillation in the whole T range measured (1.4–100 K), suggesting the formation of ‘‘[H₃O]⁺-like” H₂µ⁺O ions in the sample. Further, the absence of an oscillation in the D₂O absorbed sample also evidences the presence of H₂µ⁺O. We also measured ⁵⁹Co nuclear quadrupole resonance spectra of magnetic sample. The temperature dependence of spectra shows that there is no evidence of charge ordering and CDW ordering around magnetic transition temperature.