Don't do it.

Note this, from the article on this web site "16 ways you can be phone scammed"

"In another variation of this scam 'government officials' call to offer a 'Government Grant': 'Because you pay your income taxes on time, you have been awarded a $12,500 government grant! To get your free grant, simply give us your checking account information, and we will deposit the grant into your bank account!'  You may receive a message like this, where the caller claims to be from a government agency or some other organization with an official sounding name. The caller might claim that you will qualify to receive a "free grant" to pay for education costs, home repairs, home business expenses, or bills. In any case, the claim is the same: your application for a grant is guaranteed to be accepted, and you'll never have to pay the money back."

So they didn't say it's a government grant.  It's a giveaway for child support.  And they didn't want your personal information.  They just wanted $400 for mailing fees.

An overnight FedEx costs maybe $15, not $400.

It's a scam.  Don't call them.