It’s unlikely. Cold and flu viruses are usually species-specific, thanks to evolution. So a strain of cold or flu that could say, make your dog or cat sick won’t have the same effect in your body. (Interestingly, the reverse may not be true: Researchers from Oregon State University have identified cases in which the H1N1 flu virus, also known as swine flu, was likely transmitted to some cats, dogs, and ferrets from humans.)
In contrast, humans do appear to be able to catch the flu from pigs, which some people keep as pets. Specifically, the CDC has documented cases of a strain of influenza A (H3N2) normally found in pigs making their way to humans through contact at agricultural fairs. For that reason, the agency advises that children younger than 5 years, people 65 years and older, pregnant women, and people with certain long-term health conditions (including asthma, diabetes, heart disease, and weakened immunity) avoid pigs or swine barns.
The bottom line is that it doesn’t hurt to implement safety measures that you should be practicing anyway. Be sure to wash your hands often, especially after interacting with your pets or other animals, avoid sharing food with pets, and make sure that you and all of your loved ones (including the four-legged ones) are up to date with vaccinations.