No, water does not need to be refrigerated before putting it into a mini-fridge overnight. In fact, water itself is already at room temperature, so you can safely place it in a mini-fridge without any concerns.
Mini-fridges are designed to cool the items placed inside them. When you put water or other beverages in a mini-fridge, the cooling mechanism of the fridge will work to lower the temperature of the items inside, including the water. This ensures that the water will be chilled and ready to drink when you take it out the next day.
If you prefer your water to be colder or have ice in it, you can place the water bottle in the freezer compartment of the mini-fridge or add ice cubes before putting it in. However, it's not necessary to refrigerate the water beforehand.