If LIGO can detect mergers at billions of lightyears, i doubt they could ignore the "sound" of the NCC-1701 passing through our solar system. Proper access or not, there are enough scifi geeks with access to LIGO data that someone would get the word out.
Again, raw LIGO data is sheer noise. They have to interrogate the dataset with their hypothesized candidate signal signature, natural or otherwise, and see if it ‘responds’ with a hit. In the case of black hole mergers, they had an idea - a candidate model - of what to look for. In the case of a warp drive, could they even guess a signature to interrogate the dataset with?
And to the extent that the -1701 patrolling Mars-Venus might manage to ‘stick out like a sore thumb’, what about the relative blip of a Sh'Raan class embarking on some fresh surveying out of 40 Eridani A?