Botox has been used in ophthalmology for years for blepharospasm and strabismus; well before its cosmetic indications. Drs. Carruthers, a Vancouver couple ophthalmologist and dermatologist, are credited for the discovery of its cosmetic indications. Ophthalmologists, especially ocular plastic surgeons, have an excellent knowledge of facial anatomy. They are well qualified for multiple facial procedures.
We get this type of question often because we have nurses and physician assistants injecting Botox. The bottom line is that most intelligent and dedicated practitioners that have put much time and effort into studying facial musculature and anatomy and the effects of Botox on the facial musculature are going to be qualified to inject Botox. The question is not so much about their speciality, but about the dedication to the art and science of injection procedures. If the practitioner takes the time to get properly trained, they will have satisfied patients and word of mouth referrals will follow. While not everyone has a friend using Botox, most of us do. So try asking around to find out who your friends or relatives trust. Good luck.
The answer is - it depends. I would be hesitant about any doctor without formal training in Botox and filler injections. If the eye doctor you choose has formal training and experience in performing such injections, you should be just fine.