No. If the child is not your own child the only thing you "get" is a $500 credit for other dependents if you can claim your GF's child.
Does your GF work? Why did you need childcare? If your GF worked and needed childcare so she could work then she should be claiming her child and getting the child-related credits on her own tax return. If your GF had over $4300 of income in 2020 you cannot claim her as a dependent.
CREDIT FOR OTHER DEPENDENTS
If the child is eligible to be your dependent and lives with you then you can claim child care if otherwise eligible. The problem might be that the mother would have to be working and if she is working and has to pay taxes then the child is her dependent and not yours.