With a growing demand for mobile data usage, cellular networks are facing the challenge of severe traffic overload. Device-to-Device (D2D) multicast communication, a proximity communication technique that leverages the spatial-temporal locality of mobile data usage to achieve one-to-many simultaneous transmission, provides an efficient solution to offloading heavy traffic. However, D2D multicast communication relies on users' sharing behavior, and multicasting data incur costs such as energy, which prevents the popularity of such user-driven technique. Thus, in this paper, we study the problem of incentive design for promoting D2D multicast communication in cellular networks. Specifically, we propose a contract-based incentive mechanism to optimize the operator's expected profit from motivating D2D multicast communication for content sharing with guaranteed service quality. We consider both complete and incomplete information scenarios. The proposed mechanism can provide efficient incentives under information asymmetry by delivering contracts, which satisfy nice properties such as individual rationality and incentive compatibility. Greedy algorithms with low complexity are developed based on local optimization to obtain fast solutions for contract design. A Lagrange multiplier method based iterative algorithm that can be proved to obtain optimal contracts under information asymmetry is also proposed. Numerical results show that the proposed mechanism can handle information asymmetry better and has a better performance than linear and step pricing schemes, increasing the expected profit by up to 2.49 times and 1.8 times, respectively.