pp. Riding a four-wheel all-terrain vehicle.
Other Forms
Despite Mother Nature's cruel joke with the weather so far this spring, summer really is just around the corner. That means Albertans will soon be swimming, mountain biking, horseback riding, quadding, boating, hiking and participating in a myriad of other outdoor activities they've had to put off for what seems like forever.
—Delon Shurtz, “STARS encouraging safe summer,” Lethbridge Herald, May 24, 2011
Debris gets piled up on their engines or exhaust systems and it heats up then falls off and causes wild fires.

"We’re asking individuals, as they’re quadding, to check their vehicles and keep them clean and clear of debris.
—Dave Dormer, “Officials try to head off ATV tragedies,” Calgary Sun, May 13, 2011
2000 (earliest)
Actually when I ride some guys are in amazement because I am a quadding maniac!
—girlatvr2, “Sure girls riding ATVs,” ATVConnection.com, December 15, 2000
A much earlier (late 19th century to be inexact) sense of quadding refers to the letterpress printing process of filling up too-short lines (or sometimes adjusting spaces) with quads, which are small, square blocks of metal (quad here comes from the phrase em quadrat, which refers to the letter m).

A tip of the bike helmet to P & T Heglund for suggesting this word.