Discospondylitis is most commonly seen in medium- to giant-breed male dogs with greatest risk for dogs over 10 years of age. It is an uncommon infection in small-breed dogs and cats. To answer you directly, however, no, this inflammation/infection of the intervertebral disk and the adjacent endplates and vertebral bodies is usually caused by Staphylococcal bacteria and casual contact with the puppy won't spread these bacteria to it. There are quite a few possible original sources of the bacteria, however, and those sources might also affect a puppy - the urinary tract of the cat, infection within the heart and dental disease, for example.
Please respond with further questions or concerns if you wish.