This is the method I have used in the past for cleaning bullion coins. I imagine it would translate over.
First, soak in water with mild dish detergent. This should help loosen up and remove any grease/dirt.
Next, soak it in a very mild solution of water and lemon juice (or other citric acid source/solution). Check back on it often. The milder the solution, the safer it will be, but also the longer it will take. From what I have read, citric acid won't dissolve quartz, so it should keep the specimen intact, while helping to get rid of the staining.