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 ...