This website uses "brew": https://drinks.seriouseats.com/2013/01/how-to-make-turkish-coffee-at-home-in-an-ibrik-instructions-tips-technique-coffee-brewing.html

“To brew” is the verb for coffee, also we call the sieve things “filters” in the U.S., i believe you put the coffee grounds in the filter and pour hot water on top of it over the container (i have never made coffee but felt bad not answering).

I'm afraid by far the most common words are simply "make" and "do". "Brew" is also common (perhaps less so than for tea) but my own feeling is that we only use that when it's one of the more leisurely processes such as using a filter machine. I think with a cezve you'd be more likely to use "make" or "do". I suspect choice of word here is largely a matter of personal preference.