Yes, some granites and quartz’s contain some amounts of uranium, a radioactive element. You are safe from significant exposure to a granite that contains uranium, while it is locked in the granite and/or if it is present at very low doses within the installed materials. Your main exposure risk comes not from the Uranium but from its radioactive decay product, radon, a noble gas that also radioactively decays, emitting alpha, beta and gamma radioactivity. Radon emission varies from place to place, and from materials to material , but is commonly found in spring water and hot springs, caves, mines, petroleum products and natural gas, slate and granite stone and throughout the atmosphere. Radon decays to other less radioactive compounds, with a half life of 3.8 days and because its evolution is dependent on mass of the emitting materials, the less you have of it and the more a space is ventilated, the less exposure you would have to the radiation. The most serious exposures happen in unventilated areas like uranium mines, and basements, where radon, being heavier than air will tend to accumulate in an area with high radon evolution and confined, unventilated spaces. Where your granite countertop or quartz stone was quarried from will determine the amount of uranium and radon and your relative potential exposure levels to the emitted radioactivity from Radon. Obviously, nobody can tell you this except the manufacturer or mining company for your specific stone. As a prerequisite to purchase, ask them to provide testing date showing the radiation levels from the stone. There are tests that you can buy, or depending on where you live, can sometimes get from free, that will allow you to test the radon exposure levels in your house, or even in an open field. They are cheap and can take 2-7 days.
I think we should all be concerned about cancer, but the real problem is what exposure to potentially carcinogenic or mutagenic agents is deemed an acceptable amount. The human organism and most other biological organisms, are amazing machines, that through millions of years of evolution, have developed many redundant fail-safe mechanisms to stop diseases like cancer from killing us. Cancer, actually a family of diseases characterized by the uncontrolled growth of cells, is the result of a multiple genetic mutations to several of the fail-safe mechanisms, resulting in this uncontrolled growth. Genetic mutations happen in essentially three ways. 1) They are passed down from your parents, 2) they are the result from errors in your cellular pathways responsible for cell division and DNA replication or 3) the DNA is mutated at some point in your life, through exposure of the DNA to chemicals or radiation that directly or indirectly cause mutations. You cannot control the first two, but you can control the last one. This is your only opportunity to change how and where you live, to at least partially control your destiny.
The problem is that we live in polluted cities, drinking polluted water, breathing polluted air and eating polluted, fertilized, pesticide laden foods. Our technologies are now able to build things out of materials and chemicals that our bodies are not meant to be exposed to, like plastics, paint thinner, VOC’s, cigarette smoke and a multitude of other chemical entities. The result is that we are getting cancer at faster and faster rates because we are being exposed to DNA mutating agents at greater and greater rates and concentrations. But what has to give to be exposed to no mutagens at all. Daylight, more specifically Ultraviolet light, is a potent carcinogen, so we wear sunscreen, but did you ever notice that not everybody buys the highest rating SPF that they can get? If we really didn’t want any chance of getting skin cancer(melanoma, a particularly aggressive form of cancer), we would never go outside with exposed skin. Yet we do. How many people do you know who smoke? How many people do you know who drink alcohol, which increases the risk of over half a dozen different types of cancer? What are we willing to cut out to decrease our risks of getting cancer?
So, now back to your question. When you get the radiation information back from a manufacturer, only you can decide what is an acceptable amount of radiation to expose yourself, or your family, to. Is it none at all or is it more that absolutely nothing because you know that your house is big and you keep it well ventilated, so the exposure potential will always be very low. Now, have you tested your crawlspace, basement or house for radon emitted from the ground in your area. There is a lot more earth under your house, then quartz in your kitchen.