Yes, this can be done by changing the Layout for Tasks in the Profiles. To do this, go to Settings-->Profiles and make the change either within Base (to be applied to ALL) or make the change within a specific Profile as needed.
Furthermore, when you are within the desired area, go to the Task Item Type and click on Field Layout. Click on Edit button for Add Mode/Edit Mode.
You will see that you can specify which fields are included in the Add/Edit of the Task, also to your specific need you can see two toggles at the top... 'Always show as pop-up' and 'Name only in inline mode'. You can make the necessary adjustments to these settings and add the desired fields so that you can add the required data when adding a Task.
Sincerely,
Boris