If you can see the end of the splinter poking through the skin, there is a good possibility that you can remove it yourself quickly and easily, provided your dog will let you. Remember — safety first! Even the nicest dog can bite if they're in pain. If you attempt this, make sure you don't push the splinter farther in. Don't be afraid to trim the hair around the splinter to make it more visible. Using tweezers (the ones from first-aid kits are the best for this kind of work), firmly grasp the end of the splinter. You want to pull it out slowly and smoothly to avoid breaking off any part of the splinter under the skin. Make sure to pull in the direction of the splinter and not up and out, as this has a tendency to break the splinter. After you remove the splinter, use a cotton ball or swab to apply some dog-safe antiseptic to the wound; talk to your veterinarian to get their recommendation for the safest antispeptics for your dog.
If the splinter is completely embedded under the skin or not visible to you, do not try to extract it. Doing so may cause stress and injury to you and your pet. You can try soaking your dog's paw in a bowl filled with a warm Epsom salt solution for five to ten minutes. Doing so may soften the area and bring the splinter close enough to the surface of the skin so that you can see it and grab it. You might also want to enlist the help of a friend or family member to help hold your dog to keep them relaxed and free you up to concentrate on getting the splinter out. Again — safety first!