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, 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.

For a left proper model category in which all objects are cofibrant (or, at least, the class of weak equivalences is closed under binary coproduct), there is also an explicit construction of Karol Szumiło that yields a quasicategory. 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.

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, 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 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.