There is no direct access to the North Rim from Page by dirt, at least not easily. It would be a full day trip on a bike (running faster speeds than a SxS).
Boondock north of Page where there is BLM, nothing easily accessible south of Page in the Rez.
Lone Rock (pay to get in) gets you some beachfront camping, and a place to ride.
For free, look at 'Seismograph' directly across from Greenhaven (unincoporated establishment on the east side of 89 right on the az/ut line). There is a pullout with a half paved road heading NW. Park where it turns to dirt. From there you can head west to the old bunker, then work your way south toward Lee's Ferry to the cliffs. From there you get a birds eye of Lee's Ferry (no water access, you are ~1000' above.
Also boondock at or just east of Big Water just across the creek (road NP-230). Tons of riding at "The Moon" and this gets you to Alstrom Point overlooking Gunsight Butte (worth the trip).
For North Rim access, you can boondock just about anywhere in the Kaibab Forest. Used to hunt south of Jacob Lake. I recommend the "3 Lakes" and "Mile and a Half" area, accessed be road 212 approx 9mi south of Jacob Lake. It is cold up there now, so plan/think about camping up there. Nothing stopping you from leaving the trailer and just doing a day trip from Page, or even from Kanab/Fredonia.
I'm sure more will come to me... It's hard to beat Lone Rock (sandy), or Wahweap (hookups). Either will cost you to get into the park, but you can ride at Lone Rock.