Spoiler:No. On a 7x7 chess board, there are 25 squares of one color and 24 squares of the other color. The knight's move takes it to a square of the opposite color, so there would be 25 knights going to only 24 available squares of that color.
Correct. There is another way to show that. Here it is. Suppose that it was possible, then to each knight there corrosponds exactly one knight that will take its square. There are 24 such pairs, which leaves one knight that isn't paired up. This knight moves, and its square is left empty. Hence there is at least 1 square with 2 knights on it. Thus, it's impossible.