I will be transferred to another branch from June onward.
Means that you will be transferred from(starting) June and all the months following June.
I will be transferred to another branch from June.
Technically that the branch comes from June, but most people would understand that you mean that you will be transferred from(starting) June and all the months following June.
I will be transferred to another branch on June.
Is incorrect, you could say:
I will be transferred to another branch in June.
Which means that in June you will be transferred.
