Relativistic coupled-cluster calculation of hyperfine-structure constants of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Th</mml:mi><mml:none /><mml:mrow><mml:mn>3</mml:mn><mml:mo>+</mml:mo></mml:mrow><mml:mprescripts /><mml:none /><mml:mn>229</mml:mn></mml:mmultiscripts></mml:math> and evaluation of the electromagnetic nuclear moments of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Th</mml:mi><mml:mprescripts /><mml…

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