Qeshm Island has no public transport at all, making it a hitchhiker's dream.
Outside of the main town (also called Qeshm), traffic is very sporadic but thanks to the generosity and boundless hospitality of locals, the first, second or third car should pick you up. Make sure you have a hat, sunscreen and water as the temperature and humidity are both extreme here. Simply walk out of town to the highway--this isn't far as the towns are tiny--and start hichhiking.
Even within small towns drivers will often pick you up if you look lost and drive you a couple of blocks to your destination.
Hitching out from the town of Qeshm is more difficult as you'll be bombarded by taxis offering to give you a lift to anywhere in the island. If you do give into their offers, make sure you negotiate the price down quite a bit.
As there is no traffic and the roads are well-paved and direct, travel times are relatively short.
This part of the country is much more conservative than the rest. Despite the weather, do not wear shorts as you will be told off by a local. Also do not touch or go near women whose faces are covered. If there is only one spare seat in public, people will shuffle around so that no woman sits next to a man she doesn't already know.
Also note that much of the population speak Arabic as a first language but everyone also speaks Farsi. Basic English is spoken by very few people.