No. A real contest won’t ask for your Social Security number (SSN) over the phone (you’ll have to give it in writing).
However, if you ever win more than $600 from a contest, you will have to provide your SSN to claim the prize since its sponsor (radio station, etc.) is required to report contest winnings over $600 to the IRS. That being said, you should always be extremely careful when anyone asks for your SSN because it could be a scam. It’s important to ask why they need it, how it’ll be used, how they’ll protect it and what will happen if you don’t give it to them.