Miejsce is neuter, so the form of the adjective will be: potrzebne. And then, the 'potrzebne mi' construction takes Nominative. So the whole sentence will be "Potrzebne mi ciche miejsce" = "A quiet place is necessary to me."
Oh, okay. I wasn't sure. I was just trying to see if there was a difference between "potrzebuję" and "trzeba mi", but now you have told me that it is old-fashioned, so now I can stick with the former.