The answer to your question is yes and no. The Milky Way is not the result of the collision of two large galaxies, because then it wouldn't have this spiral structure we observe. Simulations and observations show that when two galaxies of similar mass collide, they form an elliptical galaxy without any spiral structure.
The Milky Way, which is a quite massive galaxy, is however capturing some small galaxies. Actually, there is a small dwarf galaxy that is being merged into the Milky Way right now! It is believed that many of such mergers have happened in the past, and people are trying to evaluate how many by looking at the kind of stars that we observe today in our galaxy. There are about 8 other dwarf galaxies close to the Milky Way that will at some time merge with it. And in a very long time from now, the Milky Way will collide and merge with the Andromeda galaxy, which is a galaxy very similar to ours. The result of that collision should be a large elliptical galaxy.