A living machine was a machine identified as being a lifeform, in a broader sense, including the natural acquisition of intelligence, than what was defined as an artificial lifeform.
During Commander Spock's extravehicular investigation of the inner workings of V'ger during the mid-2270s, he determined that they – the USS Enterprise and her crew – were inside of a living machine. There he saw V'ger's planet, which he described as "a planet populated by living machines," who possessed "unbelievable technology." He further elaborated that "the machine inhabiters found [V'ger] to be one of their own kind, primitive yet kindred. They discovered its simple 20th century programming: Collect all data possible."
Spock later elaborated that V'ger had become "[a] lifeform of its own, a conscious, living entity," leading Doctor Christine Chapel to inquire if he meant it was a living machine. Likewise, V'ger considered the Enterprise as a living machine, as was determined by Vice Admiral James T. Kirk, when he noted that the Ilia probe referred to the ship as an "entity". (Star Trek: The Motion Picture)
Previously the Enterprise had been recognized in the same manner, with the crew being infestations, by Nomad, and the master computer on the Shore Leave Planet. (TOS: "The Changeling"; TAS: "Once Upon a Planet")
Nearly a century later, Counselor Deanna Troi inquired to Doctor Farallon why she had so much difficulty accepting that her exocomps could be alive, noting the fact that she was, after all, talking to a living machine already, in her interactions with Data. (TNG: "The Quality of Life")