Probably not in the period of time you need it done.
You can filing a motion to withdraw your plea - I'm not saying it will be granted, but that is can be filed.
The question is do you have a case to defend? What were the facts or are you just trying to get the arrest off your record?
I suspect you didn't fight the DMV so the DMV will show a DUI is your DMV record is looked at.
Good luck with your situation.
Your time to file a notice of appeal passed 30 days after the plea.
Depending on the nature of your claim, you may still be permitted to ask the court to set aside your plea. Note that this is usually a steep hill to climb. Judges will not typically grant this motion unless you have a very good reason for doing so. The law favors "finality," and does not permit people to set aside guilty pleas based simply on "buyer's remorse" or simply regret that a different choice wasn't made.
You should speak to your public defender about your situation. Depending on the facts, that office may refer you to the alternate defender's office, which often handles defendant's claims that the public defender acted incompetently.