Sometimes things fall into place
My wife M and I have essentially fallen into a quad-type relationship without any intention of doing so. We met the other couple while we were together, we all clicked and had a great time hanging out, and we ended up getting to socialize with them without even realizing that they were poly-friendly. M realized it first and disclosed to me that she was interested in L, the male half of the couple, and that the interest was mutual. His wife S decided she was comfortable enough with the situation, as did I. Eventually S and I spent some time together to discuss the situation and get to know one another better as our spouses were beginning to explore a relationship together, and it became apparent that there was a mutual attraction between the two of us.
At this point, all four of us communicate regularly and have a significant amount of mutual respect. The relationships between M and L and between S and myself are not the same, nor could they be as we're different people, but both have evolved organically into a level of intimacy that has been both exciting and rewarding so far. As a relative newb, I'm not sure if this is a "quad" since I don't know all of the terminology, but since there are four of us that all feel as though we are in it together and are all mutually supportive and appreciative of each other, that's what it feels like to me. The relationship is still relatively new, so I can't comment on longevity or development over substantial time. All I can say is that so far, it's been an unexpected but interesting and fun experience, and I think that everyone's hope is that things continue to develop along the same lines as they have been.
So, I guess my feeling is that it is certainly possible for a quad relationship to happen naturally and spontaneously, but it depends on precisely what you are looking for, I guess. For our group, there are two established families and households and nobody is looking to change that, we are more interested in appreciating one another and helping to foster greater self-awareness and growth among everyone. Some might not consider it a "true" poly relationship for that reason, but it works for us and that's all that matters to us.