you should be able to see something in the uv spectrum, sometimes, you will have to remove the r.g,b, color filters. you can buy a monochrome sensor for that without color filters.
yes, the above method will work, remove the color filters or buy a monochrome sensor (easier), you can also put a frequency selective filter on top to limit the camera's range to the NIR spectrum you're looking for.
Is the question about the camera being able to "see" the particular wavelength? If "yes", simply look at your laser spot on the camera's screen. Most commercial cameras sensors see somewhat above 1 micron.
Or is the question about how to capture short laser flash with your camera? Dark room, the smallest diagram, the longest exposure ...