This paper studies mean-field linear-quadratic-Gaussian (LQG) games with a major agent and a large number of minor agents, where each agent's state process is driven by a Poisson random measure and independent Brownian motion. The major and minor agents were coupled via both their state dynamics as well as in their individual cost functionals. By the Nash certainty equivalence (NCE) methodology, two limiting control problems were constructed and the decentralized strategies were derived through the consistency condition. The $ϵ$-Nash equilibrium property of the obtained decentralized strategies was shown for a finite $N$ population system where $ϵ=O(1/\sqrt{N})$. A numerical example was presented to illustrate the consistency of the mean-field estimation and the impact of the population's collective behavior.