Can I export the backlog items and tasks in a sprint?

Yes. To do this, hover the mouse over the backlog items assigned to a team in a sprint, select all of those items by pressing Ctrl + A / Cmd + A, and then export them.

Alternatively, you can use filtering in the Backlog tab to show the backlog items assigned to a sprint, or a specific team in a sprint, and then export those backlog items. To do this, go to the Backlog tab and select "Filtered items" in the "Show" dropdown in the lower right corner of the page. Then filter on the sprint, and possibly a team, too, like this:

If the sprint is completed, make sure to add a Status filter criterion where you include the statuses that you are interested in (for example, "Sprint completed" and "Released"), since completed backlog items are not shown by default when you filter. Then press Ctrl + A / Cmd + A to select all of the matching backlog items, and export them.