See Converting From Any Traditional IRA Into a Roth IRA in IRS Pub 590-A:
https://www.irs.gov/publications/p590a#en_US_2021_publink1000230658
Your financial institution should have no trouble accepting this as Roth conversion (technically a conversion contribution, reportable in box 3 of the 2023 Form 5498 that they issue, not in box 1). You just need to make sure that they understand that this is a conversion contribution, not a regular contribution. If they have a deposit form for your to fill out, that form should have a way to identify the deposit as a Roth conversion. For example, the deposit slip that Fidelity provides has a box to mark that the deposit being made is a Roth Conversion, allowing the deposit to be reported in the correct box of Form 5498:
https://www.fidelity.com/bin-public/060_www_fidelity_com/documents/customer-service/deposit-slip.pdf
It's possible that the deposit slip will not even ask the date of the distribution that is being converted. It's generally your responsibility to see that the 60-day deadline is met, although some custodians do ask for the distribution date and will block the deposit unless the deadline is met (absent a reason that the deadline qualifies for an extension or a waiver).
If your financial institution is a bank, bank front office personnel are notoriously bad an understanding IRA transactions and you may have to educate them on the proper use of their own forms. Since the distribution was made from a traditional IRA, make sure that they do not try to treat it as a rollover from another Roth IRA.
