I had my IQ tested at the age of six. It was done at a school for gifted children.
Contrary to what most people think, actual IQ tests are not just about filling out a paper. Mine took a whole day and included multiple conversations with a psychologist.
I don't think that you can just go to a psychologist and ask for an IQ test simply because you are curious though. There probably needs to be a medical reason for such a test.