Calcium deficiency. Quails need access to shell grit or another calcium supplement at all times. I find shell grit pieces are far too big and need to be ground up. You can use cuttlefish bone and using the back of spoon just scrape the powder off and add it to food/water etc.
Once you sort the calcium issue, start saving your egg shells. Wash them, dry them (a few secs in microwave or oven), grind them up, and offer them in a seperate bowl for your hens -at all times. They will supplement themselves as they need.
I am new to quail (just put my first quail eggs in the incubator today), but I have seen this in my chickens before. When my hens just began laying we had a few softshell eggs. Their oviducts where getting warmed up so to speak. Their were double yolk eggs, soft shelled eggs, small eggs, extra large eggs... a lot of weird eggs. After about a week of laying, the eggs were all healthy looking. I tossed out soft shelled eggs because they lack the protective shell and could easily get infecting with pathogens. The odd shaped eggs and double yolks are totally fine to eat.
If the bird is well into laying already, I would second the thought that it is a calcium deficiency. The microwaved egg shell works great. We always put our egg shells in a bowl or cup to the side and microwave them. I then crush them with a spike meat tenderizer - a blender would work or just a zip lock bag and your hand even. We also provide oyster shell for our chickens - it is a bit large and may need additional crushing for quail - not quite certain. I do not mix the egg shell or oyster shell into the food. I just scatter the egg shell on the ground and there is an old tire laid in the run filled with oyster shell and grit. Chickens use it as they need it - I imagine quail would too.
If the problem persists after a week or so of providing a calcium source, it could be a severe calcium deficiency or a problem with the quail's oviduct. If it's a severe deficiency, try a liquid or powder calcium mixed into the feed to get level up. Oviduct problems... well that bird would be in the freezer soon. If she's a pet, let her live out her days but don't expect good eggs anymore.
None of my chookies have ever laid a soft shelled egg. But not long ago, I bought a bunch of eggs off some people who were proudly telling me they feed their chickens "only bread, lettuce and corn". Let me tell ya, the egg shells weren't 'soft' but they were as thin as paper. I did not eat or feed my doggies a single one of those eggs. I composted the lot