The answer is maybe.
If the QM instances have been created and registered correct, then the queue will be available.
You do not mention which type of QM setup you are planning to use: HA cluster, Multi-Instance, RDQM, Native HA.
If the application uses local binding then the application obviously need to be active on the server where the QM is active.
When the QM becomes active on another server (e.g. Multi-Instance) then the network settings (i.e. firewall rules) may prevent MQ client connectivity, which is why it is always a good idea to verify connectivity before real use of a QM. For that purpose the QM should be switched to become active on all the possible servers where the QM can run active and connectivity test conducted against each of the servers.