For a given integer $q$, I would like to compute the cardinality of the following set of integers: \begin{equation} N(q; \ell)=\left\{m\in\mathbb{Z}, \ell_{q-2}\ldots \ell_2\in\mathbb{Z}_{\ge 0}\, :\ |m|\le \ell_{q-2}\le \ell_{q-3}\le \ldots \le \ell_2\le \ell\right\}^{[1]}. \end{equation} Here $\ell\in\mathbb{Z}_{\ge 0}$ is fixed. I know the result and it is \begin{equation} \# N(q;\ell)=\frac{ (2\ell +q-2)\Gamma(\ell+q-2) }{ \Gamma(\ell+1)\Gamma(q-1)}=\frac{(2\ell+q-2)(\ell+q-3)!}{\ell!(q-2)!}. \end{equation} You can find it, e.g., on Müller's book on spherical harmonics. The computation you find there is based on a generating function approach: one introduces the generating function \begin{equation}\tag{1} g_{q}(x)=\sum_{j=0}^\infty x^j\cdot \#N(q; j) \end{equation} showing that it satisfies the recursive relation $g_q(x)=g_{q-1}(x)/(1-x)$. Using this one computes an explicit expression for $g_q$. Finally one computes the Taylor expansion of $g_q$ and equates coefficients with equation (1), obtaining the result.
Given the explicit nature of the set $N(q; \ell)$, I believe that a direct combinatorial argument is possible to compute its cardinality. This would avoid the use of a generating function.
[¹] The cardinality of $M(q;\ell)$ is the number of linearly independent eigenfunctions of the Laplace-Beltrami operator on the $q-1$ dimensional sphere corresponding to the eigenvalue $\ell(\ell-q-2)$. Such an eigenfunction is usually denoted by $Y_{\ell, \ell_2\ldots \ell_{q-2}}^m, $ and its parameters are called quantum numbers.