You can add a "New Task" button in the related activity section of a record's page layout by modifying the page layout.
Note: Users must have create permissions for the Task and Activity entities for the buttons to be visible.
- Navigate to Setup > Create > Entities and open the entity. In the Page Layouts tab, click the Default layout to open it.
- In the layout editor, scroll down to view the Activity History section and click the wrench icon to edit it.
- In the resulting pop-up, click the Buttons & Actions section to expand it. Here, you can click the 'New Task' button in the options list, then click the arrows to move it to the 'visible' list. Click Ok then click Save & Close.
The Activity History will now show a button to create a new task.
For more information on Entity Page Layouts, see the Help Documentation.