Of course you can do a fic trade with your friend--you don't have to go through a forum if the two if you want to exchange fics.
The way these fic trades work are equivalent to pulling names out of a hat, so to speak. You fill out the form to sign up, including fandoms that you read and write for. Then the hosting moderator will assign you a recipient. You then chose from one of the four prompts they provide and write a fic based on that. They're really fun to participate in!!
Both you and your friend are more than welcome to sign up for this fic trade as well. Just note that there's no guarantee that the two of you will receive each other as recipients.