You can modify the controller extension "PrepareProductOverviewModels". You can find it here: /Presentation/Nop.Web/Extensions/ControllerExtensions.cs.
I was hoping to do it in the view only and not recompile the source. Your post gave me the info I needed to do it that way though, so thank you. What I did (in case someone wants to do this as well):
In the _ProductBox.cshtml view I added:
Then at the end of the top
var productQuantity = EngineContext.Current.Resolve<IProductService>().GetProductById(Model.Id).StockQuantity;
Lastly, in the page I wanted to react to the stock quantity I added:
@if (productQuantity < 1)
<span class="out-of-stock">Out of Stock</span>
Thanks for the clues that helped me!