It is possible that Thranduil lived in Doriath during the War of Wrath. However it is very unlikely that he would have participated in the Battle of Unnumbered Tears or the final battle when the Valar and host finally defeat Morgoth. In the former the dwarf king kills a dragon as it crushes him. In the latter winged and non winged dragons appear as a last reserve for Morgoth and bring great destruction on the host of Valar and great eagles before they are defeated. 

So it could be that the surviving dragons of the last battle were scattered in defeat and possibly one or more of these could have ended up near Thranduil at some point.

However I think it is far more likely this is just a liberal representation in the movie of the great battles and woe experienced by the ancient elves. Just hinting to the non-read movie goers that elves had in fact fought dragons and many other horrific creatures through the millennia. Much the same way the elves showing up at the battle of Helm's Deep in the movie adaptation of _The Two Towers_ was just a representation of the elves involvement in the War of the Ring.