The best route from La Fortuna, Costa Rica to Monteverde, Costa Rica is via Route 702 and Route 1. Google/Garmin will try to take you from Route 142.
You will only encounter 10km of dirt/gravel roads. If you take Route 142 you will be driving approximately 20-25km on dirt.
Above you can see the dirt roads begin as soon as you turn right on to Route 606 and last only 10km.