Most sandstones are actually quite abundant with quartz by composition. You're probably thinking of quartzite, which is quartz sandstones which have been metamorphosed by the pressure and heat of deep burial.
Sandstone doesn't turn into quartz. Sandstone is made of indiviidual grains that usually include quartz, but also a lot of other minerals. When sandstone is weathered it breaks down into the mineral grains which are released to the environment and move away as sand. The quartz grains are more resistant to weathering and destructino than many of the other types of mineral grains, so as the sediment moves, it becomes more rich in quartz (because the other minerals are destroyed).
