In this paper, we study network games where players are involved in information aggregation processes subject to the differential privacy requirement for players' payoff functions. We propose a Laplace linear-quadratic functional perturbation mechanism, which perturbs players' payoff functions with linear-quadratic functions whose coefficients are produced from truncated Laplace distributions. For monotone games, we show that the LLQFP mechanism maintains the concavity property of the perturbed payoff functions and produces a perturbed NE whose distance from the original NE is bounded and adjustable by Laplace parameter tuning. We focus on linear-quadratic games, which is a fundamental type of network games with players' payoffs being linear-quadratic functions, and derive explicit conditions on how the LLQFP mechanism ensures differential privacy with a given privacy budget. Lastly, numerical examples are provided for the verification of the advantages of the LLQFP mechanism.