There is an analogue, but one should replace coequalizers by geometric realizations (homotopy colimits over Δop). If $F : I \to M$ is a diagram in a model category, one has
$$\operatorname{hocolim}_I F = \operatorname{hocolim}_{k \in \Delta^{\operatorname{op}}} \coprod_{i_0 \to \cdots \to i_k \in I} F(i_0).$$
For instance, see section 2 of https://emilyriehl.github.io/files/hocolimits.pdf for the simplicial model category case.
