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] V. Hinich, Homotopy coherent nerve in Deformation theory arXiv:0704.2503v4
