The movie script calls out the location as:

"EXT. SHELBY COUNTY COURTHOUSE - DAY

The old courthouse is a massive Gothic stronghold, with an armada of police cruisers parked at the curb."

https://imsdb.com/scripts/Silence-of-the-Lambs.html

However the movie locations website calls it:

"'Memphis Town Hall’, where Lecter escapes from the holding cell and borrows the face of his guard, is the Allegheny County Soldiers and Sailors Memorial Hall & Museum, 4141 Fifth Avenue, a military museum, part of the University of Pittsburgh."

http://movie-locations.com/movies/s/Silence-Of-The-Lambs.php

Shelby County Tennessee is where Memphis is.