Your broker is correct. A Roth conversion is only recognized for the year it occurs. If he converts in 2021, it cannot be treated as having occurred in 2020. It wouldn't make any sense to contribute to the Traditional and convert it to the ROTH if he is eligible for the Roth. If he makes a 2020 contribution, that would give him until the tax due date next year to make his 2021 contribution, so that would make the most sense.
To enter a 2020 Roth IRA contribution in TurboTax Desktop, with your return open, click on Search and type in "Roth IRA", then click on "Jump to roth ira". You will land on a page where you start the entry for a Traditional or Roth IRA.
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