I am looking for a reference for the maximal order of an element in PSL(2, $q$), where $q$ is prime power.

A standard reference for this type of results is Bertram Huppert: Endliche Gruppen I, Band 134 der Grundlehren der mathematischen Wissenschaften, 1967, SpringerVerlag. However I don't have that book at hand, so I cannot check. Another source you might wish to check is Jean Dieudonne: La Geometrie des Groupes Classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete 5(1963). 

