  Nuclei with a quadrupole deformation such as $^{177}$Hf have enhanced weak
quadrupole moment which induces the tensor weak electron-nucleus interaction in
atoms and molecules. Corresponding parity non-conserving (PNC) effect is
strongly enhanced in the $^3\Delta_1$ electronic state of the $^{177}$HfF$^+$
cation which has very close opposite parity levels mixed by this tensor
interaction. In the present paper we perform relativistic many-body
calculations of this PNC effect. It is shown that the tensor weak interaction
induced by the weak quadrupole moment gives the dominating contribution to the
PNC effects in $^{177}$HfF$^+$ which significantly exceeds contributions of the
vector anapole moment and the scalar weak charge. The anapole and the weak
charge can contribute due to the nonadiabatic mechanism proposed here.
Therefore, corresponding experiment will allow one to separate the tensor weak
PNC effect from the other PNC effects and to measure the quadrupole moment of
the neutron distribution which gives the dominating contribution to the weak
quadrupole moment.
