Can a laptop with 1080p native resolution output 4K signal to an external monitor?
Yes, and a longer probably Yes.
Yes #1
The output on the laptop screens completely independant of the output on attached monitors. It is no problem to have a very low resolution laptop screen (say 1920x1080) and attach a 4K monitor.
Longer Yes
The display resolution of an external monitor depends on the monitor (well duh!) and the graphics output capability of your laptop. For any modern laptop a 4K screen should not be a problem as long as you have a modern DisplayPort connector or a HDMI 2.0 connector.
Having said that. Alienware is often branded as gaming laptops. Please realise that you can perfectly attach 1 or 2 4K screens, browse the Internet, email, use spreadsheets etc etc, but that 4K gaming is still hard for most current laptops.
The laptop screen has no relationship to the laptop's output capabilities. Most laptops are designed to support external monitors for serious work. The built-in screen is often just what is practical to include for portable use within the constraints of the physical size and price point.
The output capabilities are determined mainly by the graphics card it contains. Its GPU and available memory define the upper limit of the total number of pixels it can work with. This earlier question describes how you can determine the maximum resolution the card can handle.
It isn't unusual for decent laptops to be able to work with 4K resolution, but that isn't the whole picture. The graphics card doesn't create one static image, send it to the monitor, and the work is done. For smooth video, all of the screen data is replaced many times per second (the frame rate). There is a limit to the total volume of graphics data the laptop can move.
So while it might be able to write, say, a giant spreadsheet as a 4K image to a screen, it might not be able to refresh a 4K screen fast enough for video that would be acceptable to watch.
If you're interested in video or gaming, you need to look at the laptop's capabilities in both resolution and frame rate. You might find this information in the product specs, or you can use the tools described in the linked question, above.