As Bramwell said, it’s a play on words - to say someone is ‘a bit flush’ is to say they have a lot of cash at that point, as in ‘let’s go out tonight, I’m feeling a bit flush’ i.e. ‘I’ve got some spare cash to spend’. It’s a common expression in the UK, I don’t know about anywhere else. You ‘flush’ the toilet, which describes the sudden flow of water, and also if someone is red in the face and sweaty you can say that they look ‘flushed’, as in their skin is flushed with blood.