グリーン車って英語でなんて言うの？

Any of these phrases would suffice.

First and second class are better than coach or economy classes. Sometimes they provide food and drinks too.

>Green car pass This means you have a ticket/card to enter the green car for travelling. >Business class carriage. The best available that is very comfortable and less crowded.

example "can i book a seat in first class please". or "can i book a seat in business class". or "do you have a premium option". or "is there a luxury option". Some transport networks offer business or first class or premium class options when booking a ticket. Sometimes these are referred to as luxury or First class. These tickets are usually more expensive and sometimes provide food and drinks in the package.

Business and first class carriages offer a more relaxed environment. Wider seat,reclining action and greater leg room usually. Also better service from train staff and food and drink. Premium economy if available would offer a slightly better service than a standard ticket. Beware these options cost a lot more than standard pricing