Unripe peaches will never ripen if they are stored in the fridge
There are certain fruits that just don’t like being anywhere near the cold, and unripe peaches have joined that club. Although ripe peaches will just about survive for a day or two in these conditions, unripe peaches just don’t have a chance if you decide to keep them in the refrigerator instead of on the counter.
The cold and moisture in the fridge could mean peaches stay unripe forever and leave you unable to enjoy them properly. If you want them to ripen naturally, keep them on the countertop until they start to change. If you want them to ripen quickly, stick them in a paper bag alongside a ripe banana.