In the books, Klaus's first and only confirmed kiss is with Violet Baudelaire, a kiss that was interrupted by Count Olaf's interference. This kiss is portrayed as a significant moment in their relationship, and while their feelings for each other are clear, their romantic connection is ultimately thwarted by the unfortunate circumstances of their lives.
Therefore, based on the available information, it's safe to assume that Klaus Baudelaire did not have a second kiss in the books.