  We have performed Co-nuclear quadrupole resonance (NQR) studies on
Na$_{x}$CoO$_{2}\cdot y$H$_{2}$O compounds with different Na ($x$) and hydrate
($y$) contents. Two samples with different Na contents but nearly the same
$T_c$ values ($x$ = 0.348, $T_c$ = 4.7 K ; $x$ = 0.339, $T_c$ = 4.6 K) were
investigated. The spin-lattice relaxation rate $1/T_1$ in the superconducting
(SC) and normal states is almost the same for the two samples except just above
$T_c$. NQR measurements were also performed on different-hydrate-content
samples with different $T_c$ values, which were prepared from the same
Na-content ($x$ = 0.348) sample. From measurements of $1/T_1$ using the
different-hydrate-content samples, it was found that a low-$T_c$ sample with
$T_c = 3.9$ K has a larger residual density of states (DOS) in the SC state and
a smaller increase of $1/T_1T$ just above $T_c$ than a high-$T_c$ sample with
$T_c$ = 4.7 K. The former behavior is consistent with that observed in
unconventional superconductors, and the latter suggests the relationship
between $T_c$ and the increase in DOS just above $T_c$. This increase, which is
seemingly associated with the two-dimensionality of the CoO$_2$ plane, is
considered to be one of the most important factors for the occurrence of
superconductivity.
