Q. I saw an article that I can report the expenses as a gift using the form 709. Not sure how to do this?
A. What is it you're trying to do? Asking if you "can" report it on form 709, makes it sound like you think there's some benefit to doing so. There isn't. Tax benefits for education come only with filing a regular 1040 with form 8863, but only if she is your dependent.
[EDIT] If she is not your dependent, YOU cannot claim anything for paying her tuition. She can claim a tuition credit, on her tax return, even though you paid the tuition. But, it is a non refundable credit. It will do her no good, if she doesn't have enough income to have a tax liability.
Form 709 is something totally different*. It has nothing to do with income tax and is file separately from a 1040 regular tax return.
Paying your child's (dependent or not) educational expenses, is an exception to the gift tax form reporting requirement. No form 709 is required to be filed, in your situation. See: https://www.thebalance.com/what-gifts-are-not-subject-to-the-gift-tax-3505684
*"Gift Tax" is somewhat of a misnomer. Even though a gift tax return is sometimes required (but not in your case), very few people ever actually pay federal gift tax. The purpose of the gift tax return is usually only to document a reduction in the allowable estate tax exemption.
See https://turbotax.intuit.com/tax-tools/tax-tips/Tax-Planning-and-Checklists/The-Gift-Tax-Made-Simple/...
