No, the NCLEX is a national standardized examination, not a state exam.
NCLEX=National Council Licensure Examination. That's for Practical Nurses (NCLEX-PN) and Registered Nurses (NCLEX-RN). PNs or RNs take the same examination in whatever state of residence.
Once you pass it once, you never have to take it again. If you plan on moving to another state to practice, you will do what is called, "endorse" to that state. In other words, you follow that state's requirements for endorsement (all paper work, background checks, etc.), but you do not take the NCLEX again.