As long as you keep the quail eggs in the refrigerator and there are no temperature drops such as the power going out to the refrigerator then the quail eggs can last up to 6 weeks in the refrigerator.
Quail eggs should always remain refrigerated when you're not cooking them so don't leave the quail eggs out on the counter at room temperature for too long after you cook some of them.
Put the rest of the quail eggs back into the refrigerator soon after you remove the quail eggs from the refrigerator to keep the rest of the quail eggs at a cool enough temperature.
Then after 6 weeks if they're not used up then the quail eggs are likely bad and should be thrown out.