It's not quite in the literature, but there is a fully explicit construction that avoids hammock localisation or any kind of fibrant replacement: by a [recent result of Lennart Meier](http://arxiv.org/abs/1503.02036), a certain "double cosubdivision" of the Rezk classification diagram of a model category is a complete Segal space, so (by a result of Joyal and Tierney) we can take degreewise 0-simplices to get a quasicategory. The vertices of the quasicategory constructed above are indeed all the objects of the model category we start with, and the edges are diagrams of the form
$$\bullet \rightarrow \bullet \leftarrow \bullet \rightarrow \bullet \leftarrow \bullet$$
where every arrow except possibly the interior $\rightarrow$ is a weak equivalence. Perhaps this is more complicated than you hoped for, but it is unavoidable in the general case because we do not always have a two-arrow calculus.

There is also an [explicit construction of Karol Szumiło](http://arxiv.org/abs/1411.0303) that makes a quasicategory out of a category of cofibrant objects. The vertices are cofibrant cosemisimplicial resolutions, and similarly, the edges are resolutions of cospans. The homotopical correctness of this construction was recently proved by [Chris Kapulkin and Karol Szumiło](http://arxiv.org/abs/1506.08681).