Data encryption has been widely studied in cloud data management to protect data security. Most encryption schemes only works with fully trusted cloud services and have issues with key-escrow problems, which result in high storage overhead for key certificates and the security is relatively low. To address these concerns, certificateless encryption was formalized. However, in practical cloud applications, secret key exposure threat delegates a severe and realistic concern, potentially leading to privacy disclosure. In addition, most schemes are susceptible to resist quantum computing attacks. As such, research focusing on designing a forward secure certificateless encryption scheme while enjoying quantum-safety is far-reaching. In this paper, we propose FSCE, the first lattice-based forward secure certificateless encryption scheme, which can reduce the damage from key exposure in a quantum setting. We employ the lattice basis sampling algorithms to construct the certificateless encryption. Moreover, unlike former schemes, FSCE utilizes lattice basis extension algorithm together with binary tree structure to achieve one-way secret key evolution, allowing users to update their secret keys periodically. Furthermore, our scheme is proven to be secure in IND-CPA under two types of adversaries. Comprehensive experimental evaluations indicate that FSCE is effective and practical for cloud data management.