Absolutely not. The American Veterinary Dental College has put out a position statement against anaesthetic free dental cleanings, that can be found by clicking here.
There is no way that a proper cleaning can be done without an anaesthetic. Awake dental cleanings can only remove tartar from above the gumline, as apposed to below the gumline where gingivits and periodontal disease actually occur. The results are purely cosmetic and do not help your pet’s oral health in the slightest. If an animal moves during the procedure, then there is risk of lacerating the gums, tongue or other oral structures with the sharp dental instruments used to clean teeth.
It is not safe, and it is illegal for anyone (including veterinarians) to offer anesthesia-free dentistry in Ontario.