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.
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.