Signcryption is a cryptographic scheme that achieves the functionalities of both public-key encryption and digital signatures. It is an important scheme for realizing a mechanism of sending and/or receiving messages in a secure way, since it is understood that signcryption is a public-key based protocol to realize a secure channel from an insecure channel. On the other hand, various post-quantum cryptographic schemes have been proposed so far. Recently, several cryptographic schemes have been proposed in the quantum random oracle model where an adversary can submit quantum queries to a random oracle. In this paper, we propose a generic construction of signcryption in the quantum random oracle model for the first time. Our construction achieves both of the strongest confidentiality and strongest integrity in the multi-user setting tightly.