A 2MP camera can produce a 1920×1080 image only if its actual output is at least 1920 pixels on the long side and 1080 on the short side. So in pure pixel-count terms, yes, about 2MP is enough for 1080p.

But there are important caveats:

• Aspect ratio matters. A 4:3 2MP image may be around 1600–1680 pixels wide, which is not enough for 1920×1080 without upscaling. A native 16:9 sensor/output is more suitable.
• More megapixels usually help. Capturing at a higher resolution and downsampling to 1920×1080 often gives a cleaner result because of oversampling.
• Sensor pixels are not the whole story. Most color sensors use a Bayer filter, so full-color detail is reconstructed by demosaicing rather than captured directly at every pixel.
• Overall camera quality matters too: lens sharpness, sensor quality, processing, and noise all affect whether the final image looks good.

So: a 2MP camera may be enough to make an HD-sized file, but not necessarily enough to make a high-quality HD photo. In practice, a higher-resolution camera usually gives better 1080p results.