Unfortunately not. Tolkien never really created a full family tree for Legolas. We can trace his lineage back on his father's side to his grandfather, but no further. It is possible that his wife was one of the Silvarin Elves (Thranduil being a Sindarin elf), which is made clear in in the History of Galadriel and Celeborn (in Unfinished Tales), where there's a brief account of the arrival of Thranduil's father Oropher among the Silvan Elves of Greenwood the Great. It's made obvious there that Oropher and his small band of Sindar merged themselves completely with the Silvan people of the Wood, leaving behind their Sindarin inheritance. By this time, Thranduil had already been born, and in order to link themselves to the Silvan elves, it is likely that Thranduil had been married to one of them. But this is never dwelled upon, and so is just an educated guess.
Writing my dissertation on LOTR.