I have not used gmail before but I have had success in the past with hosted solutions. Just as a few suggestions try this.
1.) From the VMail Pro PC, go to the DOS prompt and type the command - ping smtp.gmail.com - if it fails that is a smoking gun. If it passes it will give you the IP Address that the ping was returned from. Put that IP Address in that field in the control panel under "IP Office Voicemail Pro", second tab.
2.) If you have access to the customer's email accounts from their hosted providers, you must make an account for that PC, if it is making it to the SMTP server, it maybe rejecting it.
3.) One of the simple things that is missed is in the voicemail pro, open up the VMPro client, click on "Administration" & "Preferences" & "General", go to the "SMTP or MAPI" tab, I can't remember what the label says. By default it is set to "MAPI", save and make live.
4.) If you can get a copy of Microsoft office to install on the voicemail pro PC, that is a great way to manually try to send an email from that PC and get that part of the integration working.
5.) In that ICON in the Control Panel, on the second tab their is a button labeled "check" in the lower left hand corner, I have also clicked on that to "jump start" the notification process, no explanation why but it worked for me.
I am more of a telephony expert than a LAN / WAN guy, these are some things that I have done in the past which has helped me fix these peoblems. Good luck to you, let me know if any of this helped.