Yes, lumberjacking does give you a damage bonus but only when fighting with axes. The bonus is 20% of the lumberjacking skill, or 30% at GM. That is,
0.2 * LJ = Bonus%, where LJ < 100
30% where LJ = 100
Axe weapons include the following: hatchets, double axes, battle axes, axes, large battle axes, and executioners axes.
A few of the weapons that DO NOT receive the bonus are halberds, bardiches, and war axes.