Speak and talk have similar meanings. They suggest that someone is using his voice, or that two or more people are having a discussion. We can say:
speak to somebody
talk to somebody
speak to somebody about something
talk to somebody about something
How old were you when you learned to speak?
What are you talking about?
Who were you speaking to on the phone?
Who were you talking to on the phone?
I was speaking to Mark about cricket.
But we say:
speak a language NOT talk a language
talk nonsense NOT speak nonsense.
Speak to and talk to are used more often than speak with and talk with.
He speaks four languages.
Stop talking nonsense!
I was talking to Tom yesterday.
'speak to' has a sense of a more superficial exchange of words, while 'talk to' suggests some kind of in depth discussion to obtain information. So:
I spoke to this man at the bus stop today and he was telling me his mother had the same operation as you.
I need to talk to my bank manager about taking out a loan.
"I was talking to this man at the bus stop today. He's had such an interesting life. He was telling me all about his time in India when it was still part of the Empire, and.."
In your call centre:
Somebody rings and you can see from what they are saying that you need to put them through to a technical expert. You might say "You need to speak to one of our technicians. I'll put you through." Whether the technician spends some time on the phone talking with the person to get their computer up and running again is up to the technician. Otherwise, "I spoke to a technician but he couldn't help because my guarantee had expired."
Perhaps you could give some situations that arise in your call centre and people can suggest what they would say, and why.