There is not exact equivalent to that in Quarto as the overall rendering process is a bit different. Depending on what you are doing in book_start.R, pre-render could be a replacement. Difference will be that what you do in pre-render won't be kept in the R session for rendering the .qmd document.
The solution replacement would be to have a setup chunk in all document like this
```{r, setup, file = "book_start.R", include = FALSE}
```
which could added as is in all docs, or put in a separate _setup.qmd file that you could load using
However, I understand that the bookdown feature is something that some people where relying on for new_session: true configuration. An equivalent of the rendering process could be considered in the future.
See