No, as it says on their website:
Will smoking be allowed on these ships?
Subject: Smoking Policy
Azamara has two designated smoking areas. These areas are located in the aft section of the Looking Glass Lounge on the port side, and on the starboard forward section of the Pool Deck. Smoking areas have signs indicating that smoking is permitted there. All other areas of the ship are non-smoking. This includes Public Rooms, Restaurants, Pool Deck, Staterooms and Suites, Verandas and Halls.
Additionally, there is a non-smoking policy on all components of the land tour portion of all Azamara Cruises cruisetour products. For the comfort of all of our guests, we request strict adherence to these policies, and thank all of our guests for their cooperation.