This is because:
* He shares a birthday with his mother, Lily. This is established in the first book and is a crucial part of the story.
* The events of "Harry Potter and the Sorcerer's Stone" take place on his eleventh birthday. While the exact date is never stated, the characters repeatedly refer to it as "Harry's eleventh birthday", making it safe to assume it's on July 31st.
Therefore, it's highly likely that Harry's first birthday was also on July 31st.