Casting Shillelagh while holding weapon and shield

Question

I'm playing a lv4 Cleric who uses a quarterstaff and a shield.

If I choose Magic Initiate as my lv4 feat and pick the Shillelagh cantrip, will I be able to cast it on my weapon without having to free up my hand(s)?

Technically, it requires both material (which i could provide with the shield hand using the holy symbol) and somatic components, the latter requiring a free hand. That means I should drop my weapon, which makes no sense since I need to hold it ("The wood of a club or quarterstaff you are holding is imbued with nature’s power[...]").

Answer 1

Short answer: Not while your shield is donned

Here's the rest:

You could not use a holy symbol for shillelagh

Your Spellcasting Focus feature says the following:

You can use a Holy Symbol (see "Equipment") as a spellcasting focus for your cleric spells.

Shillelagh is not a cleric spell for you so a holy symbol cannot be used as a spellcasting focus, you must use the material components of...

mistletoe, a shamrock leaf, and a club or quarterstaff

You don't need a second free hand, but you do need one

Material components can be accessed with the same hand as the hand used for somatic components according to the Spellcasting rules in the Player's Handbook:

A spellcaster must have a hand free to access a spell’s material components... but it can be the same hand that he or she uses to perform somatic components.

(Emphasis mine)

What this all means for you

You require a free hand to access the material components and the only eligible target is a held quarterstaff or club. The hand holding the weapon is not considered free for the purposes of providing components. Therefore, you must doff your shield to cast shillelagh on the quarterstaff.

How to get around this

If the quarterstaff is your spellcasting focus, as would be the case if you use the magic item Ruby of the War Mage (XGtE 138 and quoted below) for example, you could cast shiellelagh including the somatic components with the hand holding the quarterstaff since the quarterstaff would then substitute for the material components.

this ... ruby allows you to use a simple ... weapon as a spellcasting focus for your spells.

Answer 2

The quarterstaff is a material component of the spell so you can use the same hand to provide somatic components for Shillelagh.

As a general rule about somatic components:

If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures.

It is then clarified that this can be the same hand that is being used to manipulate material components:

A spellcaster must have a hand free to access these components, but it can be the same hand that he or she uses to perform somatic components.

Shillelagh has the following components:

Components: V, S, M (mistletoe, a shamrock leaf, and a club or quarterstaff)

Since the quarterstaff is a material component of the spell, by the rules as written you can perform the somatic components while using the hand to hold the quarterstaff.

However...

Strictly, a cleric's holy symbol is only a focus for your cleric spells, so cannot be used to replace the other material components for this druid spell you have learned. This means you need to get some mistletoe and a shamrock into that hand as well. It's up to your GM whether they think you would have to let go of your staff briefly in order to be able to get the other items in hand, or they let you do it without needing to drop it.