It does. The cup is almost entirely water, and the minerals in it take part in extraction. Hard water dulls the flavour; water with no minerals leaves the cup hollow.
Minerals pull the aromatic compounds out of the coffee. Distilled or heavily purified water cannot do that job: even a well-made brew comes out flat and empty.
High-limescale water flattens the cup: it suppresses the acids and the coffee tastes duller than it is. The same water leaves scale inside your kettle.
A jug-style carbon filter removes the chlorine smell and makes a difference in most kitchens. If you buy bottled water, pick one with a low-to-moderate mineral content on the label, not the most mineral-rich one on the shelf.
The same grind gives a different cup on different water. Expect to look for your setting again the day you switch. That is not a mistake — it is proof the water was a variable all along.
Fix My Coffee logs every brew, so you can look back and see exactly when things started changing.