"股票经纪业务" can be translated into English as "stock brokerage business" or "equity brokerage services".
"Stock brokerage business" directly refers to the business activities involved in buying and selling stocks on behalf of clients, which is a common and straightforward translation.
"Equity brokerage services" emphasizes the services aspect of facilitating equity (stock) transactions for clients, which is also a valid and often used expression in the financial industry.
Both translations are accurate and can be used depending on the context and the specific emphasis you want to convey.