Contribution

No.  This does not effect your taxes at all and nothing needs to be entered on your taxes.  The money used to open the Roth IRA will be considered a gift to your God daughter.  A Roth IRA is funded with after-tax money, so there is no deduction.

This deposit by you on your goddaughter's behalf will be a gift of $6,000 to your goddaughter and a Roth IRA contribution by your goddaughter.  Your goddaughter must be eligible to make this contribution, otherwise the contribution is subject to an excess-contribution penalty on your goddaughter's tax return.

If your goddaughter uses TurboTax to prepare her tax return, she should enter the $6,000 contribution to  allow TurboTax to determine if she is eligible to make the contribution and to determine if this retirement contribution by her qualifies her for a Retirement Savings Contributions Credit.

If you give your goddaughter a gift, that is not deductible to you.  Whatever she does with the gift would be reported on her tax return.  She is not allowed to contribute to an IRA unless she has income earned from working that is equal to or more than the contribution. 

If she is young and not working yet, there may be better ways of investing for her future, but you would need to consult a financial advisor since I don't know all the options and rules.