I am currently using QGIS 2.18.15 and I have two sets of polygon layer data. I would like to use the Select by Location research tool in order to isolate the polygons that have their centers within the other separate layer. ArcMap has a specific "have their centroid in" option when selecting by location. I don't see that QGIS has this. Is there an equivalent?
There is no direct equivalent in Qgis core, but you can use the "Select Within" plugin, which should do exactly what you need (it adds an entry in the vector menu):
Otherwise, you would need to extract the centroid as a new layer, then make the selection on it.