Unfortunately, the answer is "maybe" and we need more information.
If the child lives in your home with your partner (the grandparent) and yourself more than half the year, and you are not married to the partner, then your partner (the child's grandparent) has the better default claim to the child as a dependent. If you have been legally appointed the child's guardian by the state, then you also have a claim. (But I wonder then, why the state didn't appoint your partner.)
In the case where both you and your unmarried partner both qualify to claim the child as a dependent, then either one of you, but not both, can claim the child. You can agree who will claim the child. If you don't agree, the IRS has certain tie breaking rules you must follow.
If other adults live in your home with the child (such as one of the child's parents) then things get more complicated.
