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Hopf insulators are exotic topological states of matter outside the standard ten-fold way classification based on discrete symmetries. Its topology is captured by an integer invariant that describes the linking structures of the Hamiltonian in the three-dimensional momentum space. In this paper, we investigate the quantum dynamics of Hopf insulators across a sudden quench and show that the quench dynamics is characterized by a $mathbb{Z}_2$ invariant $ u$ which reveals a rich interplay between quantum quench and static band topology. We construct the $mathbb{Z}_2$ topological invariant using the loop unitary operator, and prove that $ u$ relates the pre- and post-quench Hopf invariants through $ u=(mathcal{L}-mathcal{L}_0)bmod 2$. The $mathbb{Z}_2$ nature of the dynamical invariant is in sharp contrast to the $mathbb{Z}$ invariant for the quench dynamics of Chern insulators in two dimensions. The non-trivial dynamical topology is further attributed to the emergence of $pi$-defects in the phase band of the loop unitary. These $pi$-defects are generally closed curves in the momentum-time space, for example, as nodal rings carrying Hopf charge.
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