I don't know if it is called called 부산 사투리 or not but it is the way of speaking like in the movie friend(친구)
do we learn it separate like diff grammars or becoming good in korean can make me know how to speak it
yes you can call it 부산사투리.
and i don't know why you want to learn.
but it is not easy I am native korean and i have a girlfriend from Busan.
we've been together near 10 years
when I immitate 부산사투리 she always says that is totally wrong. and i think she doesn't understand every single 부산사투리 like 아저씨 아줌마's strong accent and old-fashioned words
so if you want to be a 부산사투리 speaker study it with native like actors do when they got a 부산사람 role in the movie or live longer there.