I want to know a word for a group of students who enroll in a university at the same department at the same time. I tried looking it up in google but didn't get any real results.
You might be thinking of batch or cohort.
The batch of students that enrolled this year are very naughty.
The cohort of students that graduated last year have all found jobs.
If you want to say the "the ____ of 2010", the word you're looking for is class:
a body of students or alumni whose year of graduation is the same
I would call them 'The 2010 intake'. That is the way in which schoolchildren are described in Britain and I see no reason not to use it for university students.
To answer your question (in the comments) of what to call students who entered in 2010, you can just call them "2010 enrollees".
A person who has matriculated or been registered on a list or roll, usually at a school. See http://en.wiktionary.org/wiki/matriculant
The answers so far have given the same set of words listed in similar questions What is the difference between “Class of 2004” and “Batch of 2004”? and What is the “group name” for students who entered / left university in the same year?. (Matriculant, not mentioned in either question, perhaps is an exception, although matriculating and matriculation appear in comments to them.)
Note that none of the terms proffered so far address the “at the same department” provision of the present question. Constructions like English department cohort or Math department cohort would do so, at the cost of more words. Informally, one might refer to such a group as a coterie (“An exclusive group of people, who associate closely for a common purpose; a clique”) or clique (“A small, exclusive group of individuals; cabal”). In some cases, the departmental faculty and graduate advisors may be a camarilla.
[Link sources: ELU and Wiktionary]