No, you cannot add deposits directly from the Registers screen in BQE CORE. A register allows you to view all the transactions, generally in their date order; enter new transactions like writing checks, paying bills, making deposits, etc., and manage accounts, including making adjustments, voiding transactions, and reconciling bank accounts.
To record a deposit, you should use the Deposits screen under Accounting > Deposits. From there, you can add new funds or apply existing client payments to a bank account.