In this paper, we obtain the general solution and the generalized Hyers-Ulam-Rassias stability for a following functional equation $$sumlimits_{i=1}^mf(x_i+frac{1}{m}sumlimits_{{i=1atop j{ eq}i}.}^mx_j)+f(frac{1}{m}sumlimits_{i=1}^mx_i)=2f(sumlimits_{i=1}^mx_i)$$ for a fixed positive integer m with $m;{geq};2$. This is applied to investigate derivations and their stability on proper Lie $CQ^*$-algebras. The concept of Hyers-Ulam-Rassias stability originated from the Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(1978), 297-300.