Yes, you can say "put your thinking cap(s) on". I'm not sure if it's a slightly old idiom but we certainly recognise and understand it.

And what about "To turn on the thinking machine", not referring on a super PC, but meant ironically?

I wouldn't say that it's old - fashioned. I hear people talk about putting on different caps quite frequently. I own deBono's course called 'Super Mind Pack." I've been 'thinking' about using it for some time.

It is quite an old-fashioned one. I have not really heard people use it in a long time. (Certainly don't think DeBono coined it; although he did talk about thinking hats)

Agreed. The idiom is known and occasionally used, but it's quite old-fashioned.