It really depends on the context and what you want to emphasize/which part of the sentence you want to negate.
If for example someone claimed the exam is on Friday and you wanted to object to that, you would say:
Wir haben [nicht am Freitag] das Examen, sondern am Samstag.
If there are different points on the agenda, for example the exam and a speech. And someone claimed that the exam is on friday and you onbject:
Wir haben am Freitag [nicht das Examen], sondern den Vortrag.
Wir haben [am Freitag das Examen nicht]
sounds odd at first. Without context, the "nciht" just feels misplaced. But one example where it would sound a bit better might be:
Wir haben [am Freitag das Examen nicht], es fällt aus.
But even then you would probably rather say:
Das Examen am Freitag fällt aus
not really using "nicht".
Sadly I cant tell you the exact rules about this but this is what a native speaker would likely do. Hope this helps get a kind of feeling for the use of "nicht" in this respect.