Usually, we introduce wavenumber $\textbf{q}$ by Fourier transform, for example, an operator $A_{\textbf{q}}=1/\sqrt{N}*\sum_{i}e^{i \textbf{q}\cdot \textbf{r}_{i}}A_{i}$, where $N$ is number of sites, $i$ denotes the site index. For a lattice or finite system with regular discrete $i$ (e.g. finite (spin) chain), $\textbf{r}_i$ takes integer number, then $A_\textbf{q}=A_{\textbf{q}+2\pi\hat{1}}$. So $q$ always takes value from $(0,2\pi)$.
From solid state physics textbook, $i$ component of $\textbf{q}$ takes quantized values as $2\pi n/N_i$ once we used period boundary condition, here $i$ denotes $x, y, z$. With period boundary condition, we have translation symmetry and $\textbf{q}$ is proportional to momentum.
But what if in finite size system, what are the allowed values of $q$? For example:
1) a finite chain, when the length $L$ takes finite value, such as 50.
2) a two-leg ladder, suppose there is translation symmetry along the leg direction $x$, then we have $q_x=2\pi n/N_x$, but does it make sense to define $q_y$? (On the other hand, the Brillouin zone is one dimension defined by period along $x$ direction.)
This ladder case can also be extended to bilayer case, such as bilayer square lattice. Then does it make sense to define $q_z$ along inter-layer direction?
If this $q_y$ in ladder can be defined, how to understand it? It is not the momentum anymore.