Your mother cannot claim your daughter as a dependent just because you said she could. She has to meet all the requirements for a dependent. Go to Whom May I Claim as a Dependent? on the IRS web site and answer the questions as if you are your mother to see if she can claim your daughter.
As xmasbaby0 said, you cannot find out from the IRS whether your mother received her tax refund. But why do you need to know that? Did your mother promise to give you part of her refund? That's between you and her.
And what exactly is the fraud that you think your mother has committed? If she does meet the requirements to claim your daughter as a dependent, there is no fraud. Failing to give you money that she promised to you is not fraud.
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