Can a Nikon D3000 do this? Make the lights blurry, I have a pic sample included?

That.

And how would you do that?

It's called time exposure and it's done by leaving the shutter open for 2, 3 or perhaps 5 seconds.
I did this in my office, on a tripod with a flash light - shutter open for 5 seconds.

But first you must know how to properly operate the camera.

Yes, it's called "Rear Sync Flash" or "2nd Curtain Flash"

It's done by taking a photo with a long shutter speed (to gather the light movement), and then a flash at the end (to freeze the rest of the image.)

Here's another example of it, where the photographer had a long shutter speed, moved the camera in a circular motion, and was followed by a flash at the end to freeze the lady: