The short answer is "yes" - if the IRS believes that you owe taxes for a previous year they will withhold your refund from 2012 and will apply it to the unpaid taxes that are showing on their records.
The point here, however, is that "if" - because the IRS will generally not hold back a refund unless their records show you have unpaid taxes (or if, under certain circumstances, the IRS is required to send your refund to another government agency you owe money to; however, I'm assuming that isn't the case here).
If you want to find out what the IRS records for 2010 show about you, you can order an account transcript for your 2010 tax year online here: http://www.irs.gov/Individuals/Order-a-Transcript
That transcript should show whether or not the IRS has assessed any tax against you for 2010 yet; however, unless you've received a notice from the IRS that proposes to assess that tax, or you've received a so-called notice of deficiency regarding an assessment, the odds are that the IRS has not yet gotten around to checking up on your taxes for 2010.
Keep in mind, however, that they will, and it can be harder to work with them if you need an installment payment plan if they came after you before you filed the return. The question is not "if" the IRS will come after you, the question is only "when" the IRS will come after you.
So, that means that the best thing you can do now is to get that 2010 return completed and filed. Since you're disputing the amount of income that was reported to you on that 1099-G - and you should absolutely start that process now, so you can show the IRS you're actually doing it - and if you can properly estimate how much income you actually had, you can prepare your return reporting just the income you actually received, and then you should attach a disclosure statement to your return explaining what you did, i.e., that the 1099-G reported too much income, that you are currently disputing it with the agency, and that you reported the correct amount on your return. Use a Form 8275, Disclosure Statement, which you can download here: http://www.irs.gov/pub/irs-pdf/f8275.pdf to explain what you're doing and why.
The IRS will almost certainly disagree with you at first, but this way you've got your 2010 return in, and you've got the ball rolling forward.
My answer does not constitute legal advice and may not be relied upon by anyone for any purpose and does not constitute an attorney/client relationship or an offer to form such a relationship. This disclaimer is intended to be fully compliant with the requirements of Treasury Department Circular 230 and the terms thereof are fully incorporated by reference. If you wish to consult with me please contact me at dana@nytaxcounsel or visit my website at www.nytaxcounsel.com