I came here looking for essentially an answer to the original question, but none of the replies so far actually provide a reference (HTT 4.2.4 does not contain the result, as far as I can tell.)

In [1] there is a proof that the ordinary and homotopy coherent nerve of a "fibrant groupoid" are homotopy equivalent (corollary 2.6.3). They also comment that this result can "probably" be deduced from work of Cordier-Porter, Lurie and Joyal.

Here "fibrant groupoid" means "fibrant simplicially enriched category with homotopy category a groupoid", i.e. a simplicially enriched category in which all morphism spaces are kan complexes, and all (0-)morphisms are equivalences. Given a topological group $G$, such a category is produced by taking the one-object category with mapping space the singular complex of $G$.

[1] http://arxiv.org/pdf/0704.2503.pdf