Complexity of computing the union of H-polytopes in three dimensions
Your problem is solvable in polynomial time, as the dimension is fixed (at $d=3$):

Let $h_1,\ldots,h_n$ be an enumeration of all the bounding hyperplanes of the
polytopes $P_1,\ldots,P_k$ and $Q$.
Compute the arrangement of $h_1,\ldots,h_n$ (the subdivision of three-dimensional space into vertices edges, faces, and cells). This can be done in polynomial time; see for instance &quot;Constructing Arrangements of Lines and Hyperplanes with Applications&quot; by H. Edelsbrunner, J. O’Rourke, and R. Seidel, SIAM Journal on Computing 15 (1986), pp 341–363.
Now check whether there exists some cell $C$ in the arrangement that is contained in $Q$ but in none of $P_1,\ldots,P_k$.
(For checking containment, it is enough to check whether the vertices of
the cell satisfy the inequalities of the polytope.)
If such a cell $C$ exists, answer NO. Otherwise answer YES.

If the dimension $d$ is given as part of the input, then your problem
(&quot;Does the union of $P_1,\ldots,P_k$ cover all of $Q$&quot;) is coNP-hard.