What computer are you talking about? The 737 isn't a Fly By Wire aircraft. There is no computer for the trim system.
There *is* a computer for the trim system... it provides Mach Trim, Speed Trim, and of course MCAS. These are active while the pilot is flying manually. It is called control augmentation rather than fly-by-wire because it can be overpowered/disabled by the pilots.
And I guess the altitude hold of the autopilot, which is another computer, would also want to control pitch via the trim, and therefore would be interested to know if the trim had been disabled.