I am looking for an explicit representation of the fundamental group of a closed orientable surface of genus >1. I guess they should be abundant in degree 2. Did anyone see the explicit matrix construction of such a representation? Are there any integral ones? Maybe in higher degrees?

Yes, this has been done. MR1292919 (96f:30045) Maskit, Bernard(1SUNYS) Explicit matrices for Fuchsian groups. (English summary) The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), 451–466, Contemp. Math., 169, Amer. Math. Soc., Providence, RI, 1994. You can find the paper with Google Books, if you don't have easy access to the library. 


Here are the examples from Narasimhan and Seshadri that I mentioned in my comment above. (I could have the reference wrong; I'm actually taking this out of notes from a talk I gave many years ago.) One can vary this example in a number of ways to get other interesting examples. 

