what's the right expression( or daily used) for this? thank you in advance

I mediate between people who want to get plastic surgery to clinic?

if you mean that you are the middle man between a person seeking plastic surgery and a plastic surgeon, then this one is not appropriate for the situation.

I broke people who want to get plastic surgery to clinic?

I think you mean broker. ??

I introduce people who want to get plastic surgery to clinic?

i would go with this one

or is there other right expression for this?

pls let me know