Is it possible to use a weapon as a normal weapon and an arcane focus at the same time?

I am relatively new to D&D, and am currently creating my very first PC, a Tiefling Sorcerer with a Draconic Origin. As I was working on my weapons and items, I got into what an arcane focus is and how they work.

Since I read online that a quarterstaff can double as a weapon and an arcane focus, I was wondering if it was possible to use them as both of those things at the same time?

Let me give an example: I am a 3rd level Sorcerer and know the metamagic Quickened Spell and the Shocking Grasp cantrip. I am in the middle of combat, and on my turn decide to attack an enemy within range with my quarterstaff. Before I attack, I decide to expend my two sorcery points to change the casting time of Shocking Grasp cantrip to one bonus action. I then attack and hit the monster.

Can I simultaneously hit the monster, costing 1 action, and cast my Shocking Grasp cantrip, costing 1 bonus action, to deal more damage with my attack?