Given where you are likely to be staying in Denver I'd go to Roxborough State Park. There's hiking that's probably closer to you (eg. Cherry Creek) but it's not as good IMO.

If you have spikes and poles, you might try Rocky Mountain National Park but almost all the trails will still be covered in snow in April and Trail Ridge Road will be closed.

The flatirons in Boulder might be another option (although, like RMNP, it would involve driving through Denver from Tech Center).

Garden of the Gods is very scenic but I'm not sure there's a hike that's as long as four miles there (unless you include walking along a road).