As a carpenter, I know the REAL solution to this problem from a more practical point of view.
Dress the corners on one end of your square dowel and drive the peg in the hole.
The edges of the hole will shear off the remaining edges of the square dowel.
If you're actually doing this somewhere besides a sheet of paper, you will find that
drilling a round hole is much easier than trying to chisel out a square one.
As a side note, make sure that your dowel stock is kiln dried and it will stay tight for years to come.