The entries for Staff and Quarterstaff are different in the equipment list. Under the Arcane Focus section you get:
a specially constructed staff (p. 48).
this is distinctly different (and quite a bit more expensive) than a typical quarter staff (the arcane focus version runs 5gp vs 2sp for a quarterstaff). So no, a quarterstaff cannot double for an arcane foci staff, or the other way round (at least without further clarification).
In the Druidic Focus section you get
a staff drawn whole out of a living tree (p 151 PHB)
The druid is much closer to nature and uses a druidic focus rather than an arcane one. We see that the staff for the druidic focus has a far more...utilitarian description. There is nothing here that makes it not a quarterstaff (though it may be rougher than a typical staff). My guess is most Druids would have no problem wielding their staff as a quarterstaff. I mean who is going to argue with you, you have a quarterstaff.
Last thing, there is an argument to be made for allowing implement staves to be treated as quarterstaves under the improved weapon rules on p 46 of BD&D
In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
This means that if your staff focus resembles a quarterstaff (for some value of resemble I guess), then it could be treated as one and your proficiency bonus would apply to attacks with it (and it'd get a d6 for the damage die).