Finding containing sets, within sets of sets
The problem you are trying to solve is called PARTIAL-MATCH: given a database of sets, and a query set, find all sets in the database that contain the query. 

The most relevant paper is one by Charikar, Indyk and Panigrahy from ICALP 2002: 
New Algorithms for Subset Query, Partial Match, Orthogonal Range Searching, and Related Problems

That's on the theoretical side. In practice, many simplifications can be made if one knows something about the sets and the query. For example if $|Q|$ is small, you might be able to get away with an inverted index.