There is a common misconception that Star Wars weapons can not penetrate the shields of ships from Star Trek because the shields are resistant to lasers. This point has been argued in several fanboy wars, and as far as I can see, will be argued for as long as fanboys exist. However, Star Wars weapons actually fire particle beams (1). This is confusing as the term "turbolaser" is often used to describe the weapons found on capital ships (2). Turbolasers are actually scaled-up versions of blasters and should not be confused with lasers. These weapons would be able to penetrate the shields of the Star Trek ships. Both universes consider laser technology to be outdated, just as flintlocks are considered to be outdated in our world.
(1) - http://starwars.wikia.com/wiki/Blaster
(2) - http://starwars.wikia.com/wiki/Turbolaser