Answer
Yes, $700>26*26 = 676$
Work Step by Step
There are 676 initial pairs available and 700 people. Therefore, by pigeonhole principle, at least one pair must share a birthday. This is because after assigning 1 initial pair per person for the 1st 676 in an attempt to avoid sharing initial pairs, the next person has no possible initial pair, so they must share a pair of initials with someone. The only way to avoid this is to have someone before share a pair of initials, so a pair of people must always share a birthday.