The good answer is “Pourrai-je passer l’interro de vendredi ?”.
“Puis-je passerai l’interro du vendredi ?” is not good.
Basically, you should conjugate only one verb by sentence member.
In your sentence, there is only one sentence member
so you can only conjugate “pouvoir” and “passer” must stay at the infinitive.

Although it may sound quite illogical since the quiz was never taken in the first place, most French students would say:

“Est-ce que je peux refaire ou repasser l’interro de vendredi après les cours?”

de is also better than du, unless you take a quiz every Friday.

nota: people tend to say “évaluation” ou “éval'” nowadays;; “l’interro” is slightly outdated.