My English teacher told me that "going to go" doesn't exist in English but I saw it many times. For example "The whole band is going to go to show their support". I'd like to know if it's possible to use it or not.
Yes, it's fine, providing you want it to mean what it says.
Going to... indicates a future action:
I'm going to hit him
I'm going to vomit
So you can express the future action of going somewhere:
I'm going to go somewhere
You can add the reason for that future action.
I'm going to hit him to show my disapproval.
I'm going to go to show support.