Can I complete The Final Descent without the Ancient Nordic Pickaxe?

Pickaxe from Crescius is not needed to complete this quest, but it does collect Stahlrim inside Raven Rock Mine (the place of the final descent quest).
I will still tell you how to get it in case you have not started.

You must discuss with Glover Mallory a Corberoc walker (if my memories are good he is a blacksmith). You must discuss with him about the pickaxe, he will tell you to go and force Crescius to hand it over.
The old man will not refuse you, once you have spoken to Crescius you return to Mallory who will give you as a gift the pickaxe.

As F0XS states, you do not need the Ancient Nordic Pickaxe to resolve the quest The Final Descent. However, notably, the quest Deathbrand DOES require an Ancient Nordic Pickaxe (not necessarily the one from Crescius) to resolve. Specifically:

The trip to Gyldenhul Barrow requires an Ancient Nordic Pickaxe to break through to the passageway to the treasure and the primary antagonist of the quest.