Answer
Yes. $500>366$, so by pigeonhole principle this is true.
Work Step by Step
There are 366 available birthdays for residents and 500 people. Therefore, by pigeonhole principle, at least one pair must share a birthday. This is because after assigning 1 birthday per person for the 1st 366 in an attempt to avoid sharing birthdays, the next person has no possible birthday, so they must share a birthday with someone. The only way to avoid this is to have someone before share a birthday, so a pair of people must always share a birthday.