Moving causes stress and stress tends to make them stop laying. They will usually lay the eggs that are already in production, before stopping though, hence you can get eggs for a few days after moving them.
It could also be the change in seasons that is making them stop - they need around 14 hours of light to lay an egg a day, so unless you are supplying extra light its only natural for them to stop laying in the winter. Or maybe the new cage is in a different position where they get less light.
There are many possibilities, but according to threads on here it usually takes 2-3 weeks for quail to resume laying after being moved. If they don't have 14 hours of light, I wouldn't expect them to restart before spring though.