In this work, we systematically calculate the mass spectra of the $S$-wave fully-heavy tetraquark states, $bb\overline{b}\overline{b}$, $cc\overline{c}\overline{c}$, and $bb\overline{c}\overline{c}$, in two nonrelativistic quark models. A tetraquark state may be an admixture of a $6_c-{\overline{6}}_c$ state and a ${\overline{3}}_c-3_c$ one, where $6_c-{\overline{6}}_c$(${\overline{3}}_c-3_c$) denotes the color configuration with a $6_c$ (${\overline{3}}_c$) diquark and a ${\overline{6}}_c$ ($3_c$) antidiquark. For the tetraquark states $bb\overline{b}\overline{b}$ and $cc\overline{c}\overline{c}$ with $J^{PC}=0^{++}$, the $6_c-{\overline{6}}_c$ state is lower than the ${\overline{3}}_c-3_c$ one in both the two quark models, while the order of the $bb\overline{c}\overline{c}$ states depend on models. The $6_c-{\overline{6}}_c$ and ${\overline{3}}_c-3_c$ mixing effects are induced by the hyperfine interactions between the diquark and antidiquark, while the contributions from the one-gluon-exchange Coulomb or the linear confinement potentials vanish for the $QQ{\overline{Q}}^{\prime }{\overline{Q}}^{\prime }$ system. With the couple-channel effects, we obtain the similar mass spectra. The numerical results show that the ground $QQ{\overline{Q}}^{\prime }{\overline{Q}}^{\prime }$ ($Q=b$, $c$ and $Q^{\prime }=b$, $c$) tetraquark states are located above the corresponding scattering states, which indicates that there may not exist a bound state in the scheme of the two quark models.