  We perform the four-body calculation of the hyperfine structure in the first
rotational state $J=1$ of the H$_2$, D$_2$, and HD molecules and determine the
accurate value for the deuteron electric quadrupole moment $Q_d =
0.285\,699(15)(18)$ fm$^2$ in significant disagreement with former
spectroscopic determinations. Our results for the hyperfine parameters agree
very well with the currently most accurate molecular-beam magnetic resonance
measurement performed several decades ago by N.F. Ramsey and coworkers. They
also indicate the significance of previously neglected nonadiabatic effects.
Moreover, a very good agreement with the recent calculation of $Q_d$ based on
the chiral effective field theory, although much less accurate, indicates the
importance of the spin dependence of nucleon interactions in the accurate
description of nuclei.
