The result itself seems to be due to A. Hurwitz: "Uber algebraische Gebilde mit eindeutigen Transformationen in sich," Math. Ann., 41:403–442, 1893.

At least, this is what Babai refer to on page 42 here, as well as Macbeath on page 106 here.

If you read German, you should take a look at the paper and check if this is a correct reference and if Hurwitz used arguments similar to the ones appearing in the proof of Lefschetz fixed-point theorem.

Sorry, in my earlier answer I misunderstood your question.

The result itself seems to be due to A. Hurwitz: "Uber algebraische Gebilde mit eindeutigen Transformationen in sich," Math. Ann., 41:403–442, 1893.

At least, this is what Babai refer to on page 42 here, as well as Macbeath on page 106 here.

If you read German, you should take a look at the paper and check if this is a correct reference and if Hurwitz used arguments similar to the ones appearing in the proof of Lefschetz fixed-point theorem. My guess, however, is that he used an argument similar to the one in Rita's comment.

Note also that Lefschetz in his original paper on the fixed point theorem, see page 48 here, already knew how to prove the result on conformal self-maps using his theorem. He also mentions earlier work by Kerekjarto and Birkhoff (in 2-dimensional case), so if you are very determined, you could try to dig out the earlier references.

Sorry, in my earlier answer I misunderstood your question.

See e.g. here, page 125.

The result itself seems to be due to A. Hurwitz: "Uber algebraische Gebilde mit eindeutigen Transformationen in sich," Math. Ann., 41:403–442, 1893.

At least, this is what Babai refer to on page 42 here, as well as Macbeath on page 106 here.

If you read German, you should take a look at the paper and check if this is a correct reference and if Hurwitz used arguments similar to the ones appearing in the proof of Lefschetz fixed-point theorem.

Sorry, in my earlier answer I misunderstood your question.