I transferred HSA money from my old Company HSA account to my new Company HSA account. Should I report this transfer in my 2020 tax return? If yes, how should I do it?

If you received a form 1099-SA reporting the transfer, then you would need to enter that in TurboTax so form 8889 Health Savings Accounts (HSA) could be prepared and submitted with your tax return.

Otherwise, you would not need to report it on your tax return.