Could the Death Star travel faster than light?

Yes, of course! If it could not travel faster than light, interstellar travel would be impossible, and battle station that would be restricted to a single star-system would be pretty useless.

In Star Wars Star Wars Episode IV: A New Hope, the Death Star travel travel from Alderaan to Yavin IV, which are not in the same stellar system. Without faster than light capacity, that travel would have taken years.

Here is what the first Death Star Wookiepedia article have to say about it :

The battlestation also included two massive sublight engines in the midsection, as well as a formidable hyperdrive system. Driven by 123 individual generators tied to one navigational matrix, the Class 4 hyperdrive was fast enough to allow the Death Star to travel thousands of light years from Alderaan to Yavin IV in only a few hours.