QuickOPC controls the connections automatically. For this reason, there is no specific method to connect "on demand".
It is possible to "trigger" the connection by using some workarounds. You can do some (otherwise irrelevant) operation on the server to initiate the connection. For example, you can do a "browse" (e.g. on the root level), or make a Read of an item which may even be invalid and does not correspond to specific PLC (even though the server will return an error). If the connection to the server is not just a one-time thing but a lasting requirement, you can use Subscribe instead of Read.
It should be noted, however, that one of the design ideas behind OPC is that the fact whether the server is running or not should not have any meaning or make a difference in the results (besides performance). And, if the performance related to startup of the server is a concern, the server should be kept running by some other way than from the clients (e.g. by configuring it to be an auto-started Windows service). If controlling the server lifetime from the clients is somehow necessary to get the right results, I would say that something is not completely correct in either the design of the implementation of the system.
Best regards