I have added QueryService Capability in Content Discovery. So do I need to install Experience Optimization in presentation server to fetch the ST promotion using Smart Target API?
You will need the client API in your web application, so that it can talk to the service. That is available in the folder
Experience Optimization\Content Delivery\roles\api\rest on the installation media and is referred to as "API (RESTful) Role" in the documentation.
There is a large section in the documentation explaining what needs to be installed for Experience Optimization to work, broken down into sub-sections. In this case, the relevant sub-section is "Installing Experience Optimization on the presentation environment".