While there are a few instances where specific names are used, such as "M5" (a specific model computer in Star Trek: Voyager), the most common and consistent reference is simply "the computer".
This reflects the technology of the time and the understanding of computers. In the 1960s, when the original series was made, computers were still in their early stages, large, and dedicated to specific tasks. So, the concept of a single, powerful computer that could handle all of the Enterprise's needs was more abstract than having a specific name.