So, the canonical answer is the [q-Lucas theorem,][1] as pointed out in the comments.

This was proved in
<cite authors="Olive, Gloria">_Olive, Gloria_, [**Generalized powers**](http://dx.doi.org/10.2307/2313851), Am. Math. Mon. 72, 619-627 (1965). [ZBL0215.07003](https://zbmath.org/?q=an:0215.07003).</cite>  
[1]: https://www.math.upenn.edu/~peal/polynomials/q-analogues.htm#prelimQanaloguesQLucas