Will a laser genesis facial cause eyebrow hair loss?

Any laser treatment, including Laser Genesis, can cause hair loss. However, Laser Genesis should not have been used close to your brow. With time, hopefully, the hair on your brow will return to normal. In the meantime, Latisse could possibly help.  

Laser Genesis can result in hair loss including brow hair is a YAG laser used for non-ablative skin rejuvenation and is claimed to improve skin texture, reduce wrinkles, reduce acne scars, and diminish pore size. Laser Genesis is touted as a "lunchtime procedure" by the manufacturer, Cutera.

Laser Genesis is not used for laser hair removal purposes though some more superficial and shallow hair follicles can be affected by any type of laser treatment. When you say your lower left brow hairs that leads me to believe you are talking about the area above your eyelid....which should NOT be treated with a laser of any kind. Hopefully your hairs will return to normal, but this area should not have been treated to begin with. If you want to make your brows grow faster you could try some Latisse in that area - it's an off-label use of the product but I have seen it help many people. "This answer has been solicited without seeing this patient and
cannot be held as true medical advice, but only opinion. Seek in-person
treatment with a trained medical professional for appropriate care."