Yes, in Fast Fill the "When" section gives you full control over the layout of the shifts you fill in.
To make a Q2 pattern on a different day, click on the Frequency sub-section, select Q2, and then click on a a day in the calendar that you want to be "on".
Make sure you click on the blue underline day in the calendar -> Jul 4.
Please see this page if you haven't yet, which may be helpful.
Olivier Forget
Owner and Developer
MedRez.net Scheduling Tool