Hi Mrs M,
I don't think you need to put your dog in quarantine; from elsewhere on t'interwebz, I've found the following:
1. A special Import Permit by the Ministry of Agriculture is required to enter Iceland with a pet. This permit is valid for a period of one year.
2. Your dog or cat must be microchipped with a 15 digit ISO pet microchip and vaccinated at least 30 days but not more than 12 months prior to entry to Iceland.
3. If your dog or cat is entering Iceland from a country other than Australia, Faroe Islands, Finland, Great Britain, Hawaii, Ireland, Japan, New-Zealand, Norway (Svalbard not included) or Sweden, it will need a Blood Titer Test at least 30 days after the rabies vaccination.
4. Within 10 days of travel, the Iceland Veterinary Certificate of Health and Origin is completed. Although it is not required, we recommend that a USDA or CFIA licensed veterinarian complete the form and you have it endorsed by the USDA or CFIA if traveling from the United States or Canada.
5. A copy of the Rabies Certificate should also be included.
6. A Tapeworm test is required of dogs within 10 days of arriving in Iceland. This completes your cat or dog passport to Iceland.
My experience of dogs is that how they travel always very much depends on the dog and nobody knows your dog better than you.
Good luck.