In order to clarify the carrier density dependence of the microscopic magnetic nature in Na_xCoO₂⋅yH₂O, we have performed a positive muon-spin rotation/relaxation (μ⁺SR) study of H₂O and D₂O intercalated samples. Based on the zero-field (ZF) μ⁺SR measurements, there was no clear difference between the two superconducting phases (SCI and SCII). This is also likely to exclude the exotic scenario for the superconductivity of the SCII phase, in which the superconducting state breaks the time-reversal symmetry of the Cooper pairs, resulting in the appearance of a weak internal magnetic field below T_c. Furthermore, the ZF spectrum for the H₂O intercalated 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. The absence of the oscillation in the D₂O-intercalated sample also evidences the presence of the H₂μ⁺O complex. Considering the fraction of the H₂μ⁺O signal, we have demonstrated the coexistence of [H₃O]⁺ ions and H₂O in this compound.