  $^{229}$Th is a promising candidate for developing a nuclear optical clock
and searching the new physics beyond the standard model. Accurate knowledge of
the nuclear properties of $^{229}$Th is very important. In this work, we
calculate hyperfine-structure constants for the first four states of
$^{229}$Th$^{3+}$ using the relativistic coupled-cluster method based on the
Gauss basis set. The no-pair Dirac-Coulomb-Breit Hamiltonian with the
lowest-order quantum electrodynamics (QED) correction is the starting point,
together with all linear and non-linear terms of single and double excitations
are included in coupled-cluster calculation. With the measured value of the
hyperfine-structure constants [Phys. Rev. Lett. 106. 223001(2011)], we get the
magnetic dipole moment, $\mu=0.359(9)$, and the electric quadrupole moment,
$Q=2.95(7)$, of the $^{229}$Th nucleus. Our magnetic dipole moment is perfectly
consistent with the recommended values, $\mu=0.360(7)$, from the all-order
calculation by Safronova \textit{et. al.}[Phys.Rev.A 88, 060501 (2013)], but
our electric quadrupole moment is smaller than their recommended value,
$Q=3.11(6)$, about 5\%. Our results show that the non-linear terms of single
and double excitations, which were not included in the all-order calculation by
Safronova \textit{et. al.}, are very crucial to produce a precise $Q$ value of
$^{229}$Th. Additionally, we also present magnetic octupole hyperfine-structure
constants and some important non-diagonal hyperfine transition matrix elements,
which are required for further extracting the magnetic octupole moment $\Omega$
of $^{229}$Th nucleus.
