You may use your non Quarterstaff Spellcasting Focus for spells that have BOTH Material and Somatic components, but not if it only has Somatic components.
You'll have to forgive me while I rain on a few parades of common misunderstandings.
First, to help explain the rules of the above answer, this is taken from Jeremy Crawfords Compendium of Rules Answers concerning D&D 5th Edition
((Anyway to Spoiler this long part on Material and Somatic casting?))
"...If a spell has a material component, you need to handle that component when you cast the spell (PH, 203). The same rule applies if you’re using a spellcasting focus as the material component. If a spell has a somatic component, you can use the hand that performs the somatic component to also handle the
material component. For example, a wizard who uses an orb as a spellcasting focus could hold a quarterstaff in one hand and the orb in the other, and he could cast lightning bolt by using the orb as the spell’s material component and the orb hand to perform the spell’s somatic component. Another example: a cleric’s holy symbol is emblazoned on her shield. She likes to wade into melee combat with a
mace in one hand and a shield in the other. She uses the holy symbol as her spellcasting focus, so she needs to have the shield in hand when she casts a cleric spell that has a material component. If the spell, such as aid, also has a somatic component, she can perform that component with the shield hand and keep holding the mace in the other. If the same cleric casts cure wounds, she needs to put the mace or the shield away, because that spell doesn’t have a
material component but does have a somatic component. She’s going to need a free hand to make the spell’s gestures. If she had the War Caster feat, she could ignore this restriction..."
Next, on what is and is NOT a proper RAW Spellcasting focus.
You can use a druidic focus as a spellcasting focus for your druid Spells.
A Druidic focus might be a Sprig of mistletoe or holly, a wand or
scepter made of yew or another Special wood, a staff drawn whole out
of a living tree, or a totem object incorporating feathers, fur,
bones, and teeth from sacred animals. A druid can use such an object
as a spellcasting focus.
You can use an arcane focus as a spellcasting focus for your wizard Spells.
An arcane focus is a Special item— an orb, a Crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item— designed to channel the power of arcane Spells.
An Arcane Focus Staff appears on the Adventuring Gear table and costs 5gp and weighs 4lbs.
A Druidic Focus Wooden Staff also appears on the Adventuring Gear table and costs 5gp and weighs 4lbs.
A Quarterstaff appears on the Weapons table under the Simple Melee Weapons section, costs 2 whole sp (Silver Pieces, much cheaper than gold pieces), weighs 4lbs, and counts as neither an Arcane Focus, or a Druidic Focus, since it is a weapon and only ever a weapon and not a 'Staff' especially made for the purposes of Arcane or Druid magic.
Though a few items break this rule, they are specifically called out in their descriptions, lending even more context and solidifying that the above is accurate.
Anytime a player finds a Staff magical item, such as the Staff of Fire , those staffs are considered the Arcane or Druidic versions of the spellcasting focus counterparts, not a quarterstaff, which we've concluded are two very different items.
Sometimes, a magical staff such as the Staff of Power , comes with a line in its description that says "...This staff can be wielded as a magical Quarterstaff..." and goes on to explain what benefits you gain for doing so on top of using it as a Focus for any class capable of attuning to it.
In closing: Since most magical staffs that you find can be used interchangeably by Wizards and Druids alike (they do not differentiate between a wooden druid staff and an especially made Arcane staff) , I think it is safe to assume you can use one 'staff' focus for casting both types of spells following the above rules on Material and Somatic components. Your Quarterstaff just can not double as your focus under any circumstances except when your magical staff says it can.
Interesting choice of multiclass, but the answer is still no.
Raging barbarians are subject to the following requirement as you know:
Note that it does not make exception for how you are able to cast a spell. Off the cuff spellcasting ability could stem from racial ability, class levels, taking certain feats, and having access to magic items including the aforementioned staff.
The staff gives you the ability to use your Action to cast a spell, which includes Shield (which suggests poor phrasing given that's a Reaction spell).
Regardless, your goal is to cast a spell using the staff, which is not permitted while raging.