This letter proposes a joint covert and secure transmission scheme in the non-orthogonal multiple access (NOMA) networks against internal eavesdropping and external monitoring. In the network, the strong user wiretaps the signals of the weak user while an external warden monitors the communication behaviors of the strong user. To address the issue, a random artificial noise (AN) based beamforming is adopted, which not only reduces the eavesdropping rate of the strong user but also confuses the warden. We consider the worst case, where the warden can minimize the detection error probability (DEP) by optimizing its detection threshold. Further, we characterize the closed-form expressions of the average minimum DEP of the warden, the connection outage probabilities of the NOMA users and the secrecy outage probability of the weak user under the imperfect successive interference cancellation (SIC). To improve the performance, an effective covert rate maximization problem is formulated, under the constraints of the covertness, reliability and secrecy, which is analytically solved. Numerical results validate the superiority of the proposed scheme.