Yes, you can modify these algorithms to find different shortest paths, with a few caveats.
First, there can be an exponential number of shortest paths. Consider a simple chessboard example, where you're trying to travel from the top-left corner to the bottom-right corner. Since a chessboard has 8x8 squares, you just need to travel to the right 7 times, and down 7 times, in any order. One way to find different paths is to use tie breakers that tend to produce different results. For example, you can try a tie breaker that always moves horizontally before vertically to produce one path, then one that does the opposite.
Then there's the question of how different you want the paths to be. Would you prefer paths that are as short as possible, even if they are virtually identical? Or would you rather look for paths that are as different as possible, at the cost of taking long detours?
Here's one example algorithm you can try: modifying A* so that it penalises nodes that have been used previously: http://jsfiddle.net/3huzrpyk/
This algorithm, based on this, simply looks for two paths. The second path (blue) uses a custom cost function (the g(x) part) that penalises nodes that are already used by the first path. The penalty can be customised - low values means it tries to maintain very short paths, high values (like the image) means it tries to avoid repeating nodes even if it must take a longer detour.