Yes, cats can pick up viruses from the vets office. It's happened to my cats before. One time I took all 3 of my cats in at the same time this is when I only had 3 cats. After a couple of days Storm ended up with an URI. He then gave it to both Sunny and Pepper. Pepper got worse and worse and almost died from this. They're indoor only cats and had only been to the vets so I know that this is where they picked it up. Sky has also picked up a virus at the vets before so I've had bad luck with this. I hope that you have nothing to worry about.