So, the canonical answer is the [q-Lucas theorem,][1] as pointed out in the comments.
This was proved in
Olive, Gloria, Generalized powers, Am. Math. Mon. 72, 619-627 (1965). ZBL0215.07003.
[1]: https://www.math.upenn.edu/~peal/polynomials/q-analogues.htm#prelimQanaloguesQLucas