How can I treat these bulging veins on the sides of my nose? (photo)

It’s important to distinguish between varicose veins and spider veins when choosing between laser therapy, sclerotherapy, and vein surgery. With two board certified vein specialists, our center uses a combination of visual examination, functional flow measurements, ultrasound, transillumination, and infrared light to map out vein networks prior to therapy. Lasers can be used to treat deep varicose vein insufficiency, vein perforators, and varicose veins. Sclerotherapy is ideal for spider veins and reticular veins. Whereas surgery is reserved for severely incompetent varicose vein segments. 
Dr. Karamanoukian
Realself100 Surgeon

I would suggest not to do anything about them. There is no effective treatment and with any injection/sclerotherapy you are risking the brain circulation.

I treat many types of facial veins but veins in this area I leave alone.  The reason is that the venous drainage from this area goes directly to the venous drainage (cavernous sinus) that goes to the brain.  I would not attempt to treat these veins with the possible riks of injury to the cavernous sinus.

These veins share circulation with the opthalmic vein which, as you might guess, is an important part of the circulation of your eye. Injecting this area with sclerosant may result in blindness. 
If it really bothers you that much, you *may* be able to have a surgical ligation performed after some very careful planning (Including an MRI or a CT scan to define the anatomy for your surgeon). This should only be done by an experienced oculoplastic surgeon (an opthalmologist who has done additional plastic surgery). 
If someone tells you they can handle it in the office or they can inject it or laser it - run, don't walk, out of the consultation.