Public verification enables cloud users to employ a third party auditor (TPA) to check the data integrity. However, recent breakthrough results on quantum computers indicate that applying quantum computers in clouds would be realized. A majority of existing public verification schemes are based on conventional hardness assumptions, which are vulnerable to adversaries equipped with quantum computers in the near future. Moreover, new security issues need to be solved when an original data owner is restricted or cannot access the remote cloud server flexibly. In this paper, we propose an efficient identity-based data outsourcing with public integrity verification scheme (DOPIV) in cloud storage. DOPIV is designed on lattice-based cryptography, which achieves post-quantum security. DOPIV enables an original data owner to delegate a proxy to generate the signatures of data and outsource them to the cloud server. Any TPA can perform data integrity verification efficiently on behalf of the original data owner, without retrieving the entire data set. Additionally, DOPIV possesses the advantages of being identity-based systems, avoiding complex certificate management procedures. We provide security proofs of DOPIV in the random oracle model, and conduct a comprehensive performance evaluation to show that DOPIV is more practical in post-quantum secure cloud storage systems.