I was reading up on Gegenbauer polynomials for a blog post about the kissing number problem, which I also asked a separate question about later.
On the Wolfram MathWorld plot summary for Gegenbauer polynomials, toward the bottom, there is a list of "See Also" topics. This list includes the birthday problem.
I don't see any connection between the two topics. Visiting the plot summary for the birthday problem doesn't yield anything about Gegenbauer polynomials that I can see. (As Dietrich Burde notes in the comments, however, it does link to hypergeometric functions, which in turn link to Gegenbauer polynomials.) Am I missing something, or is this perhaps a mistake?
MathWorld is Eric Weisstein's creation, hosted at the Wolfram after a lawsuit with CRC.
The link you found is probably historical. A previous version of the Birthday Problem page gave an expression for $Q_2(n,d)$ - the probability that a birthday is shared by exactly $2$ and no more people out of a group of $n$ people - in terms of an Ultraspherical Polynomial. The current Gegenbauer Polynomial page says Gegenbauer Polynomials are "are proportional to (or, depending on the normalization, equal to) the ultraspherical polynomials".
I cannot judge whether the change is a correction or a simplification, though the current Birthday Problem and Gegenbauer Polynomial pages each mention hypergeometric functions of the $\,_2F_1$ type.