The project consists on an One on One Chat System based on Sockets.
The first person sends a message to solve an issue in the company, this message is stored on the Database and waits until the second person clicks on this message to start the Chatting between them.
No offence, but this sounds a bit like "I want to build a car. It needs to have 4 weels. How can I do it with FuelPHP?".
FuelPHP is an application development framework for the PHP programming language. As such, it aids in application development (i.e. it deals with all the boilerplate like for example database access and screen generation), but it doesn't build applications for you. You have to do it yourself.
When it comes to socket communication, you're building a realtime application, for which PHP might not be entirely suitable, unless you're an expert (it requires process development, message queuing, etc).
I would more look at something like Ratcket (http://socketo.me/) to implement the communications part, and use Fuel for the interactive (web) part of it. It even uses a chatroom application as a demo.
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