In this definition BQM can be taken to be a space - the [geometric realization][1] of the [nerve][2] of the category QM. The homotopy groups are then the usual homotopy groups from topology.

There also is a definition of the homotopy group of a simplicial set - you can thus compute the homotopy group of the nerve without passing to the geometric realization first - and the definition you give looks more like that, but the answers are isomorphic.

  [1]: https://en.wikipedia.org/wiki/Simplicial_set#Geometric_realization
  [2]: https://en.wikipedia.org/wiki/Nerve_%28category_theory%29