No you can not have that removed from your record. The ony way to try to get it removed would be to file an application to withdraw your plea and if that is successful, win at trial. However, that result is very difficult to achieve even under the best of circumstances.
If this is an offense on a driver's license you currently hold, it will certainly appear. If it was obtained out of state or on an out of state license, there is a chance that it will not appear. You can always obtain your driving record (referred to as a driver's abstract) from the Motor Vehicle Commission in Trenton, New Jersey. If you did not have an attorney represent you for the DUI a lawyer can ask the court to open the matter. Then, if there is a successful defense to the charge it would be removed from your abstract. Best of Luck