Yes. BUT....
If the child is in the state's custody you will have to go through a background check much like a police officer does. You'll have to pass that and if there are any criminal charges you won't pass.
If your investigation proved to be unwarranted or the problems were solved in a good way then it shouldn't be a problem.
The good thing about adopting through the state is that they pay all costs. If that child is in their custody and the parent's rights have been terminated THEN that child is free to be adopted. The foster family will get first choice. If they want this child and they've shown he's doing well in their home then they'll get to adopt them if they want them.
If the foster family doesn't want to adopt this child they will go to the father's family and the mother's family next. If a blood relative doesn't want them then you would be the next likely person.
If the child is living at home and she just can't take care of him then the two of you can go to the courthouse, file the paperwork, you pay the thousands and thousands in court costs and attorney fees, and the child is yours. No state involvement at all.
BUT BUT BUT BUT BUT BUT.
If this adoption isn't signed off by the biological father, no matter who it is, that father can always come back and file charges against you and take this child away. He has not signed his rights away even if the mother didn't put his name in the birth certificate. If he can prove biologically that this child is his child and he wasn't consulted then that child's adoption was never valid.
You must tread lightly here. Even if the father doesn't want this child his mother and father might want their grandchild. The mother's mother or father or sibling might want this child.
You, not being related, aren't guaranteed the right to adopt until every single blood relative this child has is on board for you to adopt so they can't come back in a year, 5 years, 10 years, if you die, etc...it must be completely legally done with full representation of an attorney who is absolutely up to date on the adoption laws and that works closely with the state and how to get all parties signed off.