should be pretty simple to determine if your customer did, in fact, win the lottery, as this is public information and should be on the state lottery website. In my state, the check is drawn on a commercial bank and titled "louisiana lottery corporation" but it is not a "government" check, meaning not drawn on the state treasury, but a commercial bank.
Not all states require that winners be publicly identified.
Not all states draw checks on themselves. Many draw checks on commercial bank accounts.
Back to the OP's question, which, I believe centered on whether a check from the state's lottery is considered issued by a unit of state or local government. I am not sure where to draw the line here. I am certain that a check issued by the office of the state treasurer is issued by the state government, but I'm unsure about a check drawn on the state lottery commission or corporation. As a practical matter, I don't there's enough risk difference between the two to worry about.