  Molecular hydrogen ions are of metrological relevance due to the possibility
of precise theoretical evaluation of their spectrum and of
external-field-induced shifts. In homonuclear molecular ions the electric
dipole $E1$ transitions are strongly suppressed, and of primary laser
spectroscopy interest is the electric quadrupole ($E2$) transition spectrum. In
continuation of previous work on the H$_2^+$ ion, we report here the results of
the calculations of the hyperfine structure of the laser-induced electric
quadrupole transitions between a large set of ro-vibrational states of D$_2^+$;
the inaccuracies of previous evaluations have been corrected. The effects of
the laser polarization are studied in detail. We show that the electric
quadrupole moment of the deuteron can in principle be determined with low
fractional uncertainty $(\simeq1\times10^{-4})$ by comparing the results
presented here with future data from precision spectroscopy of D$_2^+$.
