I was taught by mentor to wear my quartz or any of my stones against my skin. But if wearing it in a watch is effective for you that's great. I am also not able to unearth my own stones. So I do buy them online or at local rock shops. But I always cleanse them before using them. How I cleanse them depneds on the stone and my intent. But in a pinch using a sage smudging stick and cleansing them on the smoke works great and I have yet to find any info that sage alters the properties of any store/crystal.
The quartz in a watch is carefully carved into specific dimensions. When electricity is applied, it vibrates at a known frequency, determined in part by its specific dimensions. This is known as piezoelectric effect. That is, it is the physical property of some substances that if it is physically manipulated in some specific manner, it will produce an electric spark -- such as used in flint-free lighters; but application of an electric current will result in oscillation of the material.
The purpose of the quartz is to set the clock frequency for the watch. I is (basically) the method for how the watch keeps time, in stead of using a tensioned spring releasing its energy through a catchment device.
Basically, the pulse from the piezo element is -- via some other circuitry -- applied to a motor which will turn in increments at the beginning of the mechanical chain.