The parity violation effects via the \mathrm5d6s\; ^3D_1 \to 6s^2 \; ^1S_0 transition have been extensively investigated in ytterbium atoms. However, the M1 transition between the excitation state \mathrm5d6s\; ^3D_1 and the ground state \mathrm6s^2 \; ^1S_0 , as well as the hyperfine-induced E2 transition, significantly affects the detection of parity violation signal. Therefore, it is imperative to obtain the accurate transition probabilities for the M1 and hyperfine-induced E2 transitions between the excitation state \mathrm 5d6s\; ^3D_1 and the ground state \mathrm6s^2\; ^1S_0 . In this work, we use the multi-configuration Dirac-Hartree-Fock theory to precisely calculate the transition probabilities for the \mathrm 5d6s \; ^3D_1 \to 6s^2 \; ^1S_0 M1 and hyperfine-induced \mathrm 5d6s \; ^3D_1,3 \to 6s^2 \; ^1S_0 E2 transitions. We extensively analyze the influences of electronic correlation effects on the transition probabilities according to our calculations. Furthermore, we analyze the influences of different perturbing states and various hyperfine interactions on the transition probabilities. The calculated hyperfine constants of the e \mathrm^3D_1,2,3 and \mathrm ^1D_2 states accord well with experimental measurements, validating the rationality of our computational model. By combining experimentally measured hyperfine constants with the theoretically derived electric field gradient of the extra nuclear electrons at the nucleus, we reevaluate the nuclear quadrupole moment of the ^173 Yb nucleus as Q = 2. 89(5) \;\rm b , showing that our result is in excellent agreement with the presently recommended value.