I was issued a check from a paid off student loan in which I overpaid interest on the loan. Do I need to claim that amount $113.65 anywhere on my tax return? If so where?

You need to claim the $114, as income, only if you previously deducted it on your tax returns. 

On the IRS forms, it goes on line 8 of Schedule 1, as other income.

In TurboTax, 

enter at:
- Federal Taxes tab (Personal in  Home & Business)

 - Wages & Income

Scroll down to:

-Less Common Income

      -Misc Income, 1099-A, 1099-C

       - On the next screen, choose  - reimbursed deductions from a prior year

            - On the next screen, choose other itemized deductions (even though the student loan interest deduction is not technically an itemized deduction

                 - On the next screen, enter your amount ($114)