Does TurboTax keep record of when a electric tax return is deposited into a bank account. If so I need to know the exact date my 2017 tax return was deposited. Thanks!

Did you file the 2017 return last year or more recently?    In any case, the official posting date of a refund direct deposit would be the one determined by your bank.   Are you unable to review your bank statement or ask your bank's customer service?

TurboTax does not get any feedback from the IRS once it receives your efiled return for processing.   The IRS might be able to say what date they sent the funds, but would probably not be aware of the receiving bank's time to receive and process (they say to allow 1-5 days after they send the funds due to variations among banks).

A third-party bank can be involved if you chose to pay any fees out of your Federal refund.   In that circumstance, the funds go from the IRS to the third-party bank where the fees are subtracted.  Then that intermediary bank sends the rest of the funds on to your bank account in a second direct deposit.   If you used that service, the intermediary company (SBTPG) that handles it might be able to tell you when they sent the funds to your bank, but may or may not know the date your bank received and processed it.

Are you able to say why you are in need of the exact date?   This is a public Internet forum, so don't post any personal info.