In Russian, the period of time when students need to pass the exams after a semester are called "sessiya"(сессия), "session". Can the same word "session" be used in English or is it just wrong?
I've not seen "session" used for this precise meaning.
A session can be "A period devoted to a particular activity." It could be used, for example as "I'm going to have a study session on Saturday". I'd probably understand "The exam session starts in May"
The expressions that are more common are "season", or using the verb "sit"
The exam season runs from May 15th to June 20th.
I'll sit the maths exam on the 14th of June.
Students will be sitting exams throughout the last term.
Note that in English schools and universities, exams are sat during the semester, usually in the last few weeks before the end of the summer term.
Ditto James K, but let me add:
You can use the word "session" in the general sense to refer to the time when students take exams, just as you can use it to refer to almost any time period devoted to a particular meeting or activity. "We will have a session for exams in late February", or similarly, "We will have an exam session in late February." "We had a recording session at the music studio." "When you have finished your session on the computer, before to log off." Etc.
But the word "session" does not imply anything about school or exams to American English speakers. (I can only address the US here, I'm not sure about other countries.) If you just said, "I'm going to a session next week", someone listening would wonder, "What kind of session? A session doing what?"