The terahertz (THz) regime still lacks a source capable of emitting single photons deterministically. Here, we propose an on-demand THz single-photon source, triggered by a sequence of coherent optical pulses used to dress a polar quantum emitter. The permanent dipole moment of the emitter activates THz transitions within its Rabi-split doublets, which are enhanced through resonant coupling to a THz cavity. We identify a challenge unique to this platform: spontaneous emission of visible photons competes with the THz process, degrading performance in two distinct ways depending on the sign of the drive detuning—either increasing the vacuum component, reducing brightness, or increasing multiphoton contributions, reducing purity. We demonstrate how to overcome those challenges through optimized pulse areas and a sufficiently high Purcell rate. A hybrid resonator-nanoparticle design for a cavity satisfying the latter requirement is presented, yielding efficiencies of 65-92% and purities of 88-100%, comparable to the best visible single-photon sources, while providing broad tunability across the THz range. Finally, we exploit the visible emission itself, using its detection to fully reconstruct the THz photon-number distribution and herald a single THz photon, boosting efficiency to 90-95%. All these capabilities illustrate the new technological opportunities unlocked by the unique integration of terahertz and visible frequencies.