No. TurboTax does not receive any type of information concerning the status of your tax refund once the tax return has been Accepted by the IRS or a State.
Neither the IRS or a State will inform you when the tax refund has been deposited. You can check the status of your tax refund using the IRS or State websites to find out if the refund has been Approved and Issued.
Use the federal tax refund website to check the refund status - https://www.irs.gov/refunds
Use this TurboTax support FAQ to check the state tax refund status - https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund
