Prominent veins on temples
While there is no recommended treatment for the veins on the temples, the appearance can be improved by injection of a hyaluronic filler such as Juvederm Ultra under the skin, around the veins. Please consult an experienced board-certified cosmetic dermatologist.
Temporal arteries and veins.
Temporal arteries pulsate and veins do not. A duplex ultrasound will distinguish artery from vein. If it is an artery I would not recommend removing it. If it is a vein then it can be treated by sclerotherapy or removal through tiny incisions (microphlebectomy).
Dupplex ultrasound can give the exact answer if that is vein or artery (if it is pulsating most probably thats a branch of temporal artery).
I dont believe if there is a safe solution to remove artery in that area. In case of vein, you can vein surgeon for sclerotherapy or microphlebectomy. Both workwell when done by experinced specilait.