In this paper, a novel method is proposed to generate high-quality chaotic signals using three cascade-coupled nano-lasers (NLs), and generate high-speed, high-quality random number sequences. The master NL (MNL) is subjected to optical feedback, and its output is injected into the intermediate NL (INL), which is further injected into the slave NL (SNL). In the simulation, we first adjust proper parameters so that the SNL generates a chaotic signal with a large bandwidth and time-delay signature concealment. After performing self-delayed differential processing on the chaotic signal, and utilizing an 8-bit analog-to-digital converter for sampling and quantization, followed by extraction of the m-bits least significant bit, the final random bit sequence is obtained. Finally, we use NIST SP 800-22 to test the generated random numbers. The results demonstrate that the obtained random number sequence successfully passed all tests specified by NIST SP 800-22.