From what I see in E6420 specifications and replacement displays at Parts-People, that laptop did not have a FHD (1920x1080) option. As a motherboard is designed to work with only a small number of compatible displays, it's unlikely that that display would work.  No, if your system does not support a FHD display, there's nothing you can do to increase the resolution of a compatible display. Edit: I also found an old Dell feature sheet on that system, confirming that there was no FHD option. These were the displays that were available: Your only option for higher resolution would be to use an external monitor.
Can I put the Latitude E5490 1920x1080 14" screen inside my Latitude E6420? If not, is there any alternative things I can do to try and ramp it up to 1920x1080?
Screen Upgrade for Latitude E6420