These two terms get mixed up constantly when people search for a studio, and they mean completely different things. Get the wrong one and you'll book a studio that can't do what you need.
A green screen is a solid green background used for chroma key work, where the green is removed in post and replaced with different footage or a different background. It's for video production, virtual sets, and compositing, not for still photography backdrops in the usual sense. Very few Berlin rental studios keep a permanent green screen set up, because most photography clients want a white or grey infinity cyclorama instead. On Berlin Studio Map, QRE8.STUDIO lists a permanent greenscreen set among its studio options. If chroma key is non-negotiable, contact the studio directly and confirm before booking.
A green room, in the traditional theatre and TV sense, is a private room for talent to wait, change, or get ready away from the set. It has nothing to do with the colour green. 3 studios on Berlin Studio Map have one of these, useful for shoots with wardrobe changes, clients sitting in, or several people cycling through hair and makeup between setups.
Need an actual seamless backdrop instead, in white, black, or colour? Berlin has 7 studios with an infinity cyclorama, listed here.