Funds earned and spent before opening a dedicated business account should be classified according to their origination.
For example, if your business received income, where did that money go? If you took the money personally, it would be considered either a 'distribution' or a 'loan' to you. It is up to you which of the two options you choose.
On the flip side, if your business had an expense that you paid personally, that would be considered either a 'contribution of capital' or a 'loan' from you.
If you choose to record these transactions as loans, you can offset them together, so you don't need two separate accounts, loan to you and loan from you.
When the bank account was opened, the initial deposit came from where? If it came from your personal funds, then it is either a 'contribution of capital' or a 'loan' from you.
From the sound of your question, you deposited what remained after the preceding income/expenses. This would, in effect, return the 'loan' account back to zero, if choosing that route.
The above would also be how to record any expenses you may pay personally for the business (if any) in the future.
Because these transactions were not through a dedicated business bank account, you can't record them in Quickbooks as checks and deposits. Instead, you can use Journal Entries. For any income received, you would debit your capital/loan account and credit your income account. For any expenses, you would debit the appropriate expense account and credit your distribution/loan account.
Also, if setting up a loan account, you should choose either Current Asset or Current Liability type. The capital contribution and distribution account should be Equity type.
Hope this helps!