No. Job expenses, including education, are no longer deductible, starting in 2018, under the new tax law.
But if the $4500 was included as taxable income on her 2018 (or earlier year) W-2, she may take a deduction (or credit) for "repayments" on your 2019 tax return.
Repayments. This situation is described in IRS Publication 525. pgs 34-35. If the amount is more than $3000, you can take a itemized deduction, on line 16 of schedule A or you can take a credit. The credit is computed by refiguring the tax return from the previous year as if the income had not been received. Then the difference in tax is claimed as a credit on the current year's return. In the forms mode (the forms mode is not available in the on-line versions of TurboTax[TT]), you can use the line 74 smart work sheet to enter the credit amount on line 74; select item D, claim of right under IRC 1341 for repayments. TT will enter "I.R.C.1341" on the line next to box d on line 74 of form 1040 (schedule 5). TT does not do the credit calculation or compare it to the alternate deduction.
So, the taxpayer has the option of either claiming the credit or deducting the repayment as a miscellaneous itemized deduction, whichever provides the greater benefit. But, you cannot file an amended prior year return.
If the amount is $3,000 or less, no deduction or credit is allowed. In the past, a miscellaneous itemized deduction, subject to the 2% of AGI threshold, was allowed. But, all miscellaneous itemized deduction, subject to the 2% of AGI rule go away in 2018. I've seen nothing yet to indicate if there will be an exception for repayments of income
