How will a still image from a Nikon d90 look on a 4K television?

Will the image take up the whole screen without loss in resolution?

Yes.

The D90 has approximately 4K in horizontal resolution and 3K in vertical resolution.
A 16:9 4K screen has a little bit less vertical resolution due to its different aspect ratio.

So a shot from a D90 that has a bit cropped at the top or bottom should fit nicely without any loss of quality.

Plenty.

The maximum 4K resolution is 12megapixels, although most are closer to 7.

The d90 produces 12mp images as a standard size, so it will look fine.

Do 4K TV's even exist yet? I thought the first models were still 6 months out.