If one thinks of homotopy groups built from the $C^*$ analogue of maps $\mathbb S^1\to X$, as said above, one can develop a theory. But such theory will be empty outside of abelian $C^*$--algebras. Consider, as an example, that $X$ may have no classical points and therefore a theory of loops with fixed endpoint will be not possible.
There are some attempts to define homotopy of NC $C^*$-algebras, basically inspired by methods developed in algebraic geometry (where rather than thinking of loops you think about coverings). The first such example I could keep track of is introduced here:
Čerin, Z. - Homotopy groups for C∗-algebras, Topology with applications, 29-45, Bolyai Soc. Math. Stud., 4, János Bolyai Math. Soc., Budapest, 1995.
and commented here: Agrigoroiaei
On the web you can also find some preprints of a different attempt, e.g. here
The fact that such attempts haven't had much followers may indicate they are (still) not quite successfull.
If such a generalization exists it should be quite nontrivial. The quantum torus, having no classical points and a classical analogue with nontrivial homotopy, should be the first test case to be analyzed.