Pick your pivot element and partition the strings into <, =, and > based on the first character only. Note this differs from classic quicksort in that we're maintaining three regions, not two. Now recurse for all three regions, except for the = region you look at the second character only. etc.
It's probably a loss for filenames which tend to be short, but for long strings this is a very easy to implement speedup. There's lots more optimizations possible with this technique, e.g. counting-sort style bucketing.