@BrookeGates -
adding on to what @NoahBoswell suggested, if the teachers class has students in sections by periods, then I would just set up each period with access to the quiz during their actual class time.
You did not mention anything about how the students are in the course, so I figured I would put the above information out there as well. For me at a university, I have multiple sections of the same class cross-listed into one course. that automatically creates the section setup for each course that is in the cross listed course.
In the case of high schools, I do not know if a teacher teaching Algebra I in periods 1,2,5,6 has a separate period designation for each of those periods are if all the students from the 4 periods are in the course. I would hope that the students are put into the course by period.
Here is the instructors guide page on differentiated assignments. Basically in the assign to box, delete the everyone tag that comes up by default. Then the options will show the sections/periods first followed by individual students
Since you were asking about quizzes, here is the instructors guide for differentiated assigning on Quizzes - it is the same principle as for assignments
Ron