An appeal would have to have been filed within 60 days of Judgment; the appropriate action would be with the trial court to "set aside" the Judgment. However. you are correct about the 2 year limitation for challenging paternity although there is a remote possibility that you could attempt another theory (like fraud?) but it's extremely unlikely. You would definitely need an experienced attorney - you could not attempt this on your own. The biggest problem is that you were "on notice" that this wasn't your child because "you were not even in the same county at the time of conception" - so you have no good legal excuse as to why you didn't challenge this at the time. A hard lesson learned....
In our American society, the state does not care whether you are the actual biological father of a child. As long as an order of support was entered and the state got federal child support enforcement matching funds, that's all that matters. There's even a new fiction out there in many states: it's called paternity by estoppel. That's where a man teams up with a mom who has other males' children and then is stuck paying their child support to the mother when that story ends. There is even the Trump fraud: send the lady to Donald Trump to get her pregnant and then live off the child support.
So federal money drives the train and in these hard economic times, the state needs its revenue any way it can get it: child support; child abuse/neglect; forced adoptions and so on.
For this reason, it is impossible to reverse a prior order of support. There's even the Bradley Amendment that prohibits a state judge from waiving child support arrears.
The way you were played into the child support system was you had a rebuttable presumption of paternity that you failed to rebut prior to entry of the divorce judgment. You didn't know. Now that the order was entered, you are stuck. An appellate court will have a field day sanctioning you and humiliating you for filing such an appeal. The only out is if your former wife comes to court under a child support petition and tells the court she is waiving arrears and discontinuing her claim for child support. I have had only one case like that in nine years of active practice.
This empowerment of the custodial litigant has spawned a national divorce fraud crisis. Hordes of custodial evaluators ignore the actual status of the parties and rig their evaluations for the mother to win. This is because the state assumes the father will earn more therefore paying more in child support (the state gets more enforcement funds up front). Many states will permit a reversal of custody later down the road upon bogus allegations of the kid and the mother being "conflictual". This means there is an adage here: fathers never get custody and mothers always lose it. That's national policy right now - not law, but policy.
Imputation leads to astronomical counsel fees and support arrearages that can be waived in kind via the transfer of property - even outside this country. Hence, child support is the tail that wags several dogs at once and when coupled with allegations of child abuse, wreaks havoc on divorcing families on an international basis.
Appellate courts don't care and federal courts will dismiss civil rights allegations. So, the state conspires to steal federal taxpayer funds and the fed doesn't mind. This is the situation Obama must discover and deal with if our economy is to be restored. Our men are supporting two households that are severely economically stressed. One mistake sends the father into jail for non payment of support and send two homes into foreclosure. Banks panic and cut off future loans. In these cases men don't even know their fathers because child support does not compel visitation. So men are working for nothing.
So not only are you stuck but the rest of are stuck right along with you until we vote this blight out of existence along with the politicians who enacted it.
The are certain exceptions that might apply but you will need to hire counsel to review your case, and prepare your motion to the court if appropriate.
My question to you would be why, if you lived in different counties when the child was conceived, you didn't consider this possibility initially?