If an agent does not pick up the call quick enough, it is getting pushed to their voicemail on their desk phone that the call has been pushed to - is there anyway to push those - Ring no answer calls back into the queue?

Is the agent using a phone connected to a different phone system? It should not go to voicemail if the phone is a station in PureCloud, but automatically roll back on queue.

Or, perhaps you have the Alerting Timeout on the Voice tab in the queue settings configured for a really long time, like 45 seconds? I haven't experimented with a call timeout set long enough to hit the voicemail threshhold. That setting should be around 8 - 10 seconds.

I don't think this should happen with a queue call. A direct call to an agent's number or extension will roll to voicemail if they don't answer, and if voicemail is enabled for them.

If the call was a queue call and the agent doesn't answer,  the call should go to the next available agent in the queue. I would recommend contacting Support so they can help you. 

I just tested in my own organization, and even with the Alerting Timeout set to 120 seconds I could not replicate what you are describing using PureCloud-based stations. The agent rolls to Not Responding and the call remains in the queue.

So, it sounds to me like you are set up with Remote stations and the phones are on a different system that has its own voicemail (but that's just my guess). If that is the case, make sure the Alerting Timeout on the queue is shorter than the number of seconds before the other system rolls to voicemail.

@George Ganahl  In PureCloud no, it should not happen and as you tested that doesn't happen. We have, however, seen this behavior when we integrate into existing phone system for telephony end points. Since they may have their existing voicemail setup we need to ensure that alerting timeout on ACD interactions is less that timeout on the remote system. So if the phone is set to send call to voicemail after 12 seconds, typical timeout we've seen, then PureCloud ACD alerting timeout should be set to 10 seconds.