You can certainly do it fairly efficiently if you are prepared to throw a bit of heavy machinery at it. Here is an outline.
Step 1. The algebra A generated by u and v is selfadjoint, so by the double commutant theorem it will suffice to prove that the commutant of A consists only of scalars.
Step 2. The subalgebra of A generated by u comprises all the diagonal matrices.
Step 3. A matrix that commutes with all the diagonal matrices is itself diagonal. So the commutant of A consists of diagonal matrices.
Step 4. A diagonal matrix that commutes with v must be a scalar.
Edit. Step 2 assumes that q is a primitive n'th root of unity. If not, for example of q=1, then the result is false.