The question is as in the title. 

> (What are some of the) are there any applications of Homotopical algebra (in the context of Quillen’s book “Homotopical algebra”) in better understanding (or developing new theory) of Lie groupoids?

I am reading Homotopical algebra hoping that this would help me to learn some $\infty$-category theory. Other than that, I do not have any plan. As I am familiar with Lie groupoids, I wanted to see if there is a place where these two interact in a nice way. 

(As of now), I do not have any better explanation than this to ask the question.