Your proficiency bonus is already applied to spell save DCs and spell attacks
The Basic Rules clarify, in the chapter on Spellcasting (emphasis mine):
The DC to resist one of your spells equals 8 + your spellcasting ability modifier + your proficiency bonus + any special modifiers.
Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.
Using a spellcasting focus only replaces certain material components...
A character can use a component pouch or a spellcasting focus in place of the components specified for a spell.
...so being proficient with an arcane focus has no affect on the details of spellcasting.
As an additional note, this is true not only for arcane foci that double as quarterstaffs, but also for such characters as bards of the College of Swords who gain:
If you’re proficient with a simple or martial melee weapon, you can use it as a spellcasting focus for your bard spells.