Can Botox Trigger Trigeminal Neurolgia?

It shouldn't, but unfortunately in medicine there are no guarantees (be wary if any doctor guarantees any result, even a particular response to a medication such as aspirin).  Based on the pharmacology of Botox, I would not expect it from this drug, but the simple act of injecting may be a trigger.

Treating the musculature around your eyes with cosmetic Botox shouldn't cause any issues and/or flare up with trigeminal neuralgia. If you haven't met with a reputable and well-trained and informed provider to weigh all of your options, that's a good place to start.

While I don't have any personal experience using Botox to treat TN, there are several articles suggesting that it is beneficial. For that reason I think it is unlikely that cosmetic Botox would make it worse.