By mistake the internal Active Directory domain was named as an Internet domain that the company does not own. Exchange 2010 was installed and runs fine internally. The problem is now they need to send/receive external emails. They want to keep their email accounts using the internal domain, should I rename the forest AD or do I have another options.
Obvioously you won't be able to receive email from the outside that are addressed from a domain that you don't own, but you should be able to simply setup a real domain, and configure Exchange to accept messages for the new real domain in addition to the domain you have. If someone else is really using that fake domain, then you will not be able to send email to them.
It will be important to make sure that your accounts use the real address as default so, messages sent to external parties will appear to be valid, and people can reply to you.
In the long term, you should seriously consider renaming the forest. It is really not a good idea to be using a domain, that doesn't belong to you.
It sounds like you've got a single Exchange 2010 server computer with all the roles installed there.
To get this going you'll need to configure an additional receive domain for the Customer's Internet domain name (just as @Zoredache says). You'll also need to configure a Receive connector and a Send connector to allow email to be received from the Internet and sent out to the Internet.
Finally, you'll need to modify the firewall configuration to allow inbound and outbound SMTP to/from the Exchange Server computer and, if necessary, create an MX record in the Internet DNS to get the email to route to the public IP address the Exchange Server computer is exposed via.
I agree with @Zoredache in that it's not a good idea to use someone else's domain for your AD forest. Unfortunately, domain rename is not supported after Exchange 2007 or 2010 is installed. You're stuck, now, with a migration to a new forest if you want a new name.
It is, apparently, possible to install Exchange into a resource forest, create a cross-forest trust, migrate all your mailboxes to the resource forest, then uninstall Exchange in the source forest, rename the source forest, and reinstall Exchange in the source forest and migrate everything back. That sounds like a horrible ordeal (in a production network-- fun in a lab, though) and I don't think I'd attempt it.