I own Nikon D90 with a 18-105mm kit lens. I have been looking for a suitable lens for birding with D90. According to Nikon support https://m.dpreview.com/reviews/nikon-af-s-18-300mm-3p5-5p6-vr would work with D90. But I guess 300mm is not enough for birding. So what alternative do I have?
That depends on your price range, the distance you are from the birds, and a range of other factors.
If you are looking for longer focal length (>300mm) lens, but don't want to spend a lot, then you may want to consider a different brand, like Sigma. A 500mm, 600mm, or 800mm lens from Nikon could set you back $10,000-$16,000.
You also have a choice between a zoom or prime lens. Prime (fixed focal length) lenses tend to be sharper and generally work better in lower light or for capturing rapid movement (birds in flight), as they have larger apertures. Zoom lenses are nice if you need a more versatile lens, or if you are going to be substantially different distances from your subjects.
Nikon lenses are designed for full frame (35mm) or cropped sensors. The D90 uses a "cropped" image sensor. You can still use a full frame lens with a cropped sensor camera, however, the image will appear to be zoomed in. For example if you purchase a 300mm lens for use with your D90, it will give the same field of view as a 450mm lens would on a FF camera, because the image sensor is smaller. Nikon calls lenses designed for full frame cameras "FX" lenses and lenses that are designed for cropped sensors "DX" lenses. You can calculate the FF equivalent field of view for a lens on a DX camera by multiplying the lens' focal length by 1.5 (e.g., 300mm lens * 1.5x crop factor = FF 450mm equivalent FoV)
Unless you have exceptionally quick fingers, you will want an auto focus lens. The D90 does have an auto focus motor in its body, so you can select from a wider range of lenses. Your camera will be able to autofocus with "AF", "AF-D", "AF-S", "AF-I", and "G" lenses. I would get a lens with "VR", as vibration reduction is particularly critical when shooting with longer focal lengths.
In summary: It depends on your budget, off brand lenses are cheaper. A longer focal length is better, but more expensive. Autofocus is a must. A wider aperture will let you use a higher shutter speed to capture motion.