I don't know off the top of my head whether it will physically fit, but I doubt it. Even if it did though, it would be unusable as the hood would be visible in the picture. (The cone of light coming into a 24-105 lens is much narrower than that coming into a 10-22 lens, so the lens hood can afford to be - and is - deeper.)
Do you need one? Define "need". The 10-22 hood is pretty shallow and doesn't offer much protection against unwanted light, and it doesn't offer much physical protection either. Whether the protetcion it does offer is worth having (and paying for) is a decision we can't make for you.