"股票经纪业务" can be translated into English as "stock brokerage business" or "equity brokerage business".
"Stock brokerage" specifically refers to the business of buying and selling stocks on behalf of clients, which aligns closely with the meaning of "股票经纪业务" in Chinese.
"Equity brokerage" is a more general term that encompasses the brokerage of various equity instruments, including stocks, and can also be used to describe this type of business.
In most contexts, "stock brokerage business" would be the more precise and commonly used translation.