Terahertz communications has been foreseen as a key enabler to the sixth generation (6G) of wireless communications systems. However, the design of spectrally-efficient waveforms and modulation schemes is an ongoing challenge in this regime. In this paper, we propose and experimentally demonstrate the transmission of M-ary carrierless amplitude and phase (CAP) modulated signals using a 253 GHz photo-mixing-based terahertz wireless communications system combined with the optical transmission over a 10-km standard single mode fiber (SSMF). Experimental results show that the CAP modulation technique has a capability to support the high-speed transmission of terahertz signals over a wide range of data rates from 4 Gbit/s to 96 Gbit/s based solely on optical intensity modulation at the transmitter side and terahertz envelope detection at the receiver side. Consequently, M-ary CAP can potentially be adopted as a low-complexity waveform and modulation contender to simplify the transceivers architectures for the sixth generation (6G) terahertz communications systems, without sacrificing the high throughput targeted by these systems.