NO, you do not use equity for income producing activities
Day trading, if this is your business that is one thing, if this is personal then it does not belong in a business accounting file. Assuming it is your business ...
QB can only handle the gross accounting, it can NOT handle stock ticker symbols, gain and loss, and will not provide anything close to a report that can be used for income tax reporting.
In truth, day trading is not something that can be done very easily in QB at all, but
1. create a bank type account for the brokerage
2. when a sales is complete
2a. post a check, do not print, from the brokerage bank and use COS, a cost of sales account you create (COGS type and COS detail) , as the expense
2b. show a deposit in the brokerage bank in the amount received, and use trading income as the source account for the deposit.
income will be reduced by expense on the P&L