I am using the ShieldUI ASP.NET charting component. I am trying to build a single chart, which repopulates with data, when you select a data point – and loads a new screen with related data. For example -> quarter selection -> data for the selected quarter.
Any pointers in the right direction are appreciated.

The scenario which you describe is supported by the ShieldUI ASP.NET chart component.
This drill-down behavior is achieved with some additional code-behind and is easy to build.
One such implementation is demonstrated in the following example:
Drill Down Example

Another example, which is a variation of the setup, which you commented on is this one:
Related Charts using drill down