For a group $G,$ the generating graph of $G,$ denoted by $Γ(G).$ We define $Q_n=\langle x,y: x^{2n}=y^4=1, x^n=y^2,y^{-1}xy=x^{-1}\rangle,$ the dicyclic group of order $4n.$ This paper primarily delves into exploring the graph characteristics and spectral properties of various matrices associated with $Γ(Q_n)$. Specifically, we determine the complete spectrum of the adjacency, Laplacian, distance, and eccentricity matrices. Additionally, we completely determine the spectrum pertaining to the distance and eccentricity matrices of the dihedral group of order $2n$, denoted as $D_n$.