Using Kostant's weight multiplicity formula, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root $μ$ in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$, which we denote $L(\tildeα)$, where $\tildeα$ is the highest root of $\mathfrak{sl}_{r+1}(\mathbb{C})$. We prove that the number of terms contributing a nonzero value in the multiplicity of the positive root $μ=α_i+α_{i+1}+\cdots+α_j$ with $1\leq i\leq j\leq r$ in $L(\tildeα)$ is given by the product $F_{i}\cdot F_{r-j+1}$, where $F_n$ is the $n^{\text{th}}$ Fibonacci number. Using this result, we show that the $q$-multiplicity of the positive root $μ=α_i+α_{i+1}+\cdots+α_j$ with $1\leq i\leq j\leq r$ in the representation $L(\tildeα)$ is precisely $q^{r-h(μ)}$, where $h(μ)=j-i+1$ is the height of the positive root $μ$. Setting $q=1$ recovers the known result that the multiplicity of a positive root in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$ is one.