Hi Paula,
Thanks for contacting us!
Unfortunately there is not an out of the box way to add a date picker in the ticket fieldsets. However, using the custom fields and some JavaScript, you could add this functionality to tickets. We do have a list of recommended developers who could help you out with the code if you need it.
If this is something you want to see implemented in future releases, I would recommend casting a vote in our Feature Ideas forum to let us know.
Let us know if you need anything else and have a great day!
Patricia