Is it known whether counting $q$-dimensional $p$-matching is $\#W[1]$-Hard?
Our recent paper shows that counting k-matchings is #W[1]-hard even in bipartite graphs. This answers your question.

Radu Curticapean, Dániel Marx: Complexity of counting subgraphs: only the boundedness of the vertex-cover number counts. CoRR abs/1407.2929 (2014)