Grant-free random access is promising for massive connectivity with sporadic transmissions in massive machine type communications (mMTC), where the hand-shaking between the access point (AP) and users is skipped, leading to high access efficiency. In grant-free random access, the AP needs to identify the active users and perform channel estimation and signal detection. Conventionally, pilot signals are required for the AP to achieve user activity detection and channel estimation before active user signal detection, which may still result in substantial overhead and latency. In this paper, to further reduce the overhead and latency, we explore the problem of grant-free random access without the use of pilot signals in a millimeter wave (mmWave) multiple input and multiple output (MIMO) system, where the AP performs blind joint user activity detection, channel estimation and signal detection (UACESD). We show that the blind joint UACESD can be formulated as a constrained composite matrix factorization problem, which can be solved by exploiting the structures of the channel matrix and signal matrix. Leveraging our recently developed unitary approximate message passing based matrix factorization (UAMP-MF) algorithm, we design a message passing based Bayesian algorithm to solve the blind joint UACESD problem. Extensive simulation results demonstrate the effectiveness of the blind grant-free random access scheme.