I don't have Maya, but I may be able to help.
I will just describe the steps the way I understand them. I am assuming you want to create a custom animation.
1. Load the G6 character. Some characters have built-in animation clips, combined in a "Persona". You want to get rid of those to avoid confusion. If there is a Persona, you will see "Perform" and "Move" options if you right-click the character. If so, select "Persona" and the "Remove Persona".
2. With your "clean" character, complete your animation.
3. With the character selected, click "Edit in 3DXchange" to load the character in 3DX.
4. Open the timeline for the character, and make sure the Collect Clip track is open.
5. Make sure to highlight Collect Clip for the length of your animation. Then right-click Collect Clip, and select "Add Motion to 3DXchange". This will bring the motion into 3DX.
6. In 3DX, in the Modify panel under Animation, notice that a clip is listed (called "motion0", or something).
7. Select this clip, and then click "Add to Perform". This is essential, otherwise you will not see the motion in the FBX export.
8. Export the character to FBX. You should now be able to select the "Include Animation" option.