There are really two questions here, or at least two different contexts.
- Is it normal to expect a quiet atmosphere when taking an exam?
- Is it normal to expect a quiet atmosphere when programming at work?
I'll try to answer both based solely on my own experience (20 years so far in IT):
The Exam Experience
As for the first, exams (of any kind) are generally expected to be taken in a quiet, peaceful atmosphere. In the situation you describe, it would be perfectly normal to speak with the exam administrator or supervisor in charge of the test, and mention that the other exam-takers are causing a disruptive environment for you. It would normally be the responsibility of that person to provide an appropriate testing environment. For most exams I've taken, such as certification exams, people were expected to leave the room before even speaking, and everything was dead quiet.
The Workplace Experience
As for the second, the actual workplace is a totally different situation. Throughout your career, you will see the full range of environments - everything from having to program at a desk on a noisy plant floor, to a cubicle in a heavily-trafficked corridor, to a quiet little office with the door closed.
My first job as a server admin, in fact, had me sometimes sitting in a room with only a thin wall separating us from the very loud, very noisy plant floor. It was enough that I had to wear headphones just to hear myself think. So that sort of situation DOES happen on occasion.
At most of my programming-related jobs, though, I've been in a room of varying size, generally with other programmers or people doing similar tasks, with each of us sitting at a desk in a short cubicle. Within the U.S., this seems to be the norm for any low or mid-level IT position. Once you get higher up the food chain (so to speak), like a senior developer or manager, then you could reasonably expect to have your own office with a door. At some companies, even a mid-level programmer could have their own office, but that's been rare in my experience.
This normal, cubicle-based arrangement is generally a quiet atmosphere, but even then there will usually be discussions going on about code, projects, and even non-work-related topics.
You may find such "background noise" to be distracting at first, but learning to work despite such distraction is key to becoming a good programmer. Requesting absolute silence in the workplace will only make you look bad and possibly alienate your co-workers. Just as a blacksmith toughens up his/her hands by exposing them to heat, your mind will eventually learn to block out the noise and concentrate despite it. And you'll be a more valuable asset for having that ability, as well.