If your vet is close and understanding, try bringing your dog to the vet just for a few social visits. Also at home make sure you are handling your dog all over, like a massage so it gets used to all the body sensations, separate and handle it's toes and paws too, it may need to get nails clipped one day.
This is the approach I mostly utilized with my 3 GSDs over the years...find those moments when your pooch is mellowed out and massage the dog from the tip of their nose to the end of their tail and down each leg to their paws. Long gentle strokes and all the while reassuring the dog with calming vocalizations...if the dog gets annoyed then stop..too little is much better than too much. After each session, you might treat them as well. Eventually, little by little..you will win them over. I have found over the years that once your dog is comfortable with you petting the soft of their stomach..you have won their confidence a 100%...something about that region on a dog has an effect on them....but it just can't be forced. A down/stay with a roll over for a "belly whomp" as I call them becomes a treat for them and certainly helps at the vet. During this process, I eventually get to their mouths, gums and teeth...but it all happens in due time.
I've always clipped my dog's nails and never had a problem and I believe this happens because of the processes I employed above....over time. Grooming the dog is another wonderful time to make this "connection" with your dog....slow, calm and again...in the beginning...too little is much better than too much.
There's a lot to be said for a solid down/stay with the dog on it's side...Over time, I get a bit more picky when the dog is in this position, so even if the dog raises her head, I'll give a gentle verbal of "down" and expect them to rest their head on the ground....and then proceed.
Patience, soothing and gentleness will win the day in this arena....at least it has worked for me.
There are so many benefits of a dog letting you handle them all over....looking for ticks, scrapes and of course clipping their nails as the previous poster cited.
Oh, even though I have reservations about bringing a healthy dog into the vet's office..I did as the other poster mentioned...took many a trip to the vet's office just to weigh her or have the staff scratch my pup's ears.