  HfF$^+$ cation is a very promising system to search for the electron electric
dipole moment (EDM), and corresponding experiment is carried out by E. Cornell
group. Here we theoretically investigate the cation to search for another
T,P-odd effect -- the nuclear magnetic quadrpole moment (MQM) interaction with
electrons. We report the first accurate ab initio relativistic electronic
structure calculations of the molecular parameter $W_M$=0.494
$\frac{10^{33}\mathrm{Hz}}{e~{\rm cm}^2}$ that is required to interpret the
experimental data in terms of the MQM of Hf nucleus. For this we have
implemented and applied the combined Dirac-Coulomb(-Gaunt) and relativistic
effective core potential approaches to treat electron correlation effects from
all of the electrons and to take into account high-order correlation effects
using the coupled cluster method with single, double, triple and noniterative
quadruple cluster amplitudes, CCSDT(Q). We discuss interpretation of the MQM
effect in terms of the strength constants of T,P-odd nuclear forces, proton and
neutron EDM, QCD parameter $\theta$ and quark chromo-EDM.
