Oblivious transfer (OT) is a fundamental cryptographic primitive. It is developed for the efficient and feasible implementation of most advanced cryptographic tasks. Today, most of the existing OT protocols’ security is based on number-theoretic assumptions. However, many number-theoretical problems are solvable by a quantum computer in polynomial time. Therefore, OT protocols with post-quantum cryptography approach are required. Multivariate cryptographic constructions are one of the potential candidates for post-quantum cryptographyas they are speedy and require only modest computational resources. This paper presents constructions of OT protocols utilizing multivariate public key cryptography (MPKC). Security of our schemes is achieved under the hardness of multivariate quadratic (MQ) problem. To the best of our knowledge, our designs are the first MPKC-based post-quantum OT protocols.