What to do if we got full cutody of my daughter 2 years ago when her mother was facing incarceration and now her mother is trying to make contact?

The custody orders that gave you full custody of your daughter two years ago will control what you have to do. 

If the orders strip her completely of any visitation rights, then you do not have to allow her access. 

If the orders say that she is to have "reasonable access" at your discretion (and no set time frames), then you need to arrange for some type of access to avoid an enforcement action.  Considering that the child probably doesn't know her, you would be reasonable in setting up supervised visitations in order to help reintroduce your daughter to her mother. 

If the orders have standardized periods of visitations (like every other Saturday), then you are required to let her have the child on those scheduled visitation days.

If your order does require either "reasonable" or standardized visits, and she is demanding to enforce those rights, you need to comply with the order or.....if you are concerned about the effect on your daughter of sudden reintegration, you can petition the court to modify the visitation orders.  Depending on how you feel, you can either ask for complete termination of her rights, limited visits, or supervised visits until she reaches a certain age.  Courts tend to be reluctant to terminate rights.  But is she has gone an extended period of time without making any contact (even a simple letter from prison) and without providing any type of support (especially now that she's on parole), you would have a better position to argue for termination.  If your orders don't give her visitation rights, you do not need to worry about a modification.

As far as contact by mother....Her mother can try to make contact as long as the custody order says that she can't.