This seems to be done in Section 6 of the very Korkmaz paper you mention, no?

In Section 6 of the Korkmaz paper he uses the Wajnryb presentation to derive a presentation on his two generators, for genus $g>2.$ For genus $2$ one can use the exact same method, but applied to Birman-Hilden's (1971) presentation of the mapping class group of genus $2.$