No, it needs to be depicted. The story is basically chronicling from a young age to adulthood. When it begins, they will be 12, when it ends, I don't know, but certainly in their 30's. The Academy Arc will need to be shown. Skipping it as flashbacks would screw up the whole thing since important events happen during as much as before and after.
Okay, so I'm noticing some conflicting info here. While you say important events happen here.
If it was in a series of say....10 hypothetical volumes, it would only encompass 1 volume, and perhaps a few chapters of the next. It's important for its position of events in the story, but it's not the most important part, it's just part of a timeline of story arcs in succession. Certain characters get introduced and actions and events that occur later have some of their origins in this arc.
You say here it's just an introduction to certain motivations.
The issue here is that what CharlesEBrown said is your best solution here. If it's only to introduce people, theres no point in making it an arc of its own and not something the characters occasionally talk about or is espoused upon.
The issue with having a complete arc is that an arc needs the proper framework of a story. You need the Beginning(entering the academy), Rising action (?), Climax (?), Resolution (?), and then the falling action (graduation)
Unless you have some central conflict during the academy arc that grows the characters in a meaningful way, there's not really much point to detailing it in your story if its only purpose is to introduce characters. This can be as simple as the MC struggling in an academy setting, to evil dark overlord #922 attempting to rule the school.
There are, however, alternatives to the flashbacks. One of which I like to call the 'Asian dinner friends.' Basically, it's common in Eastern novels (and I think a few of their cultures) that when old friends gather in one spot and eat, they recollect on their school days. This usually involves teasing eachother about the younger versions of eachother, talking about hopes and dreams they once had, what they do now, and just talking about important past events.
Another, more hated method is the monologue. Where the characters unnecessarily tell us their motivations.
If you find your central conflict, then I'd suggest keeping notes about several important factors.
-teachers.
-classes.
-important faculty that aren't teachers.
-important students.
-how grades work.
-how does one graduate.
-extracurriculars.
-student housing & care
Not necessarily in that order, and not all of it needs to be conveyed to the readers. But those basics, even with only a vague idea, should turn into an academy that feels similar to a real school.