You need to know not only what was mined in the quarry originally but also, and possibly more importantly, what it was mined from, the "country rock" as it's called in geochemistry. If you were for example mining Titanium then it's likely that the country rock is granite and there is likely to be Radium in the rocks, the water will be mildly radioactive and have dissolved radioactive Radon gas. If on the other hand you were mining metamorphic volcanic hard rock for road aggregate like they do in Northland then quarry contamination could include Sulfur, Iron and Asbestos. Lets not talk about opencast Lead or Cinnabar mines, you'd have to be extra stupid to use one of those for drinking water. Other potential rock source contaminates include Copper, Manganese, Lime, Salt, Silica, Antimony, Arsenic, and dissolved Halogens, plus rock dust and other potentially harmful particulate.
That's contamination from existing material, then you have to consider where the water is coming from. Groundwater will likely share the same contaminants as the existing rocks but may also bring in added contaminants from the same list but other areas. Surface runoff from the local area may be contaminated by fertilisers, organic material and microorganisms. Direct rainfall is likely to start off reasonably contaminant free in open country but could include acids, other industrial air pollutants in the form of Halogens, organic toxins or particulate like soot or Zinc dust from powder coating.
Then you have to think about what happens to the water that is in storage, if turn over is low, AKA you are using the ex-quarry for emergency rather than routine supply, then Stagnation is the biggest issue, stagnant water is a breeding ground for any number of micro- and macro-organisms that you don't want; many of these concentrate toxic compounds especially sulfurous contaminants. Evaporation will also concentrate any contaminants left in the water that isn't being used regularly. The other issue is that quarries are usually much deeper than they are wide which leads to generally poor circulation at depth meaning an Anoxic layer generally builds up at depth this leads to a build up of highly toxic Hydrogen Sulfide, which can lead to lethal out-gassing either on site or further down the treatment chain.
So those are the problems, the solutions go a bit like this:
chemical treatment to precipitate dissolved chemical contaminants, this turns dissolved toxic elements into fine powders that settle to the bottom of the water column. To do this you need to know what you're precipitating and how much of it, this requires careful monitoring of water conditions on the surface and at depth. The actual precipitation is usually done after the water has been removed from the reservoir.
bottom sourcing, take the water you remove from the quarry from the lowest point in the system, this improves aeration of the whole helping to prevent anoxia and stagnation.
aeration, pumping air through the water column causes most dissolved gases to be released from solution, it will also oxidise and precipitate metals not previously precipitated. This step should be conducted either in the open air or in a sealed environment where the gas coming out of solution can be contained as it can be highly toxic. Ozone is often used to kill microorganisms at this stage also.
add Flocculents, you may need several different ones to do a staged treatment, flocculation removes suspended particulate from the water, as the dissolved toxins are now particulate precipitates this removes them from the water altogether. Filtration is suggested as a final aid to this process as well as removing larger material.
additives like Chlorine can be used to add to the transport lifetime of the resulting water by reducing bacterial growth and killing existing microorganisms.
As a note you can still dam a river on a relatively flat plain, you built a large earthwork similar in construction to an Earthen Levee that raises the banks of a river and creates both a working head of water and an impoundment space behind the dam for water to pool. This is extremely labour intensive compared to a dam in hard rock at a convenient fall in the mountains but has the advantage of being put where you need it most.