If I understand correctly are you asking if Weather shows up on an ATC PSR/SSR display?
If so the answer is - not normally but sometimes if the weather is really bad then it may deflect the RADAR signal and appear as a Primary RADAR Trace.
I do understand though that there are some airports that have installed a Weather RADAR head to specifically detect weather but I dont know whether this is overlaid onto the PSR/SSR display or on a separate screen.