They are both correct. Which do you want to emphasize? That
your age will be 18? Or that you will become 18 by changing from
17 to 18?
This is not a question of correctness, but a question of what is more
idiomatic, that is, a question of how people usually say it.
Most people usually say
I will be 18 next year, emphasizing what
the age will be. (The change from the previous age is not so
important for most people.) So, to match your usage to that of
most English speakers, use
be, not
become, when talking about age.
CJ