At the citizenship stage, they ask you for paystubs, tax returns, w2, etc. As well, they can track your social security number. They have computer network system. They have great intelligence as a part of Homeland Security.
Why do you ask those questions? Please listen to lawyers' advice to work for at least 6 months to be safe. Green card is not everything, just an authorized document card to allow you to live and work permanently. If it is employment category, you must have a bona fide intent with your employer so it won't be fraud. If you are only laid off, it won't be a problem because it is not your fault. If you quit, that could be a problem. Please don't listen to friends or know that people quit like that. They could be in trouble in the future or not. Just be safe and be in a good moral character. I stayed with my employer for more than 1 year after I got my green card and I am still with my employer (despite that economy goes up or down). I will quit much later. If it is family category (marriage category), it is very risky because they have conditions (temp gc and bona fide evidences for permanent residency).
so it is up to you.
Hope this helps.