I would not attach them to the quads. If things go south for any reason, they need to be able to get away from it just the same as you.
I had a Jack Russel that liked to ride on dirt/street bikes. The dirt bike had a smaller fuel tank so I took a small plastic bucket and flipped it upside down, with a hole in the 'bottom', and used the filler cap to hold the modified bucket in place. The bottom of the bucket now facing upward gave him a flat spot to place his front legs and then he stood/sat on the seat. On faster runs I would use an animal harness and attach him to me in case we hit a bump or quick maneuver for road debris so I didn't lose him at 70+ on the interstate.
2005 2500 Cummins/48RE/3.73, QCLB, 4wd, BigHorn, Edge Juice w/ CTS + Turbo Timer,Transgo Shift Kit ISSPro Oil and LP pressure gauges, GDP 20/2 filters, Custom Diesel Steering Box Brace
'10 Forest River Shockwave Toy Hauler 21'
Honda EU3000I Genny