The following is an except from an Esri technical article:
Relating tables, via relationship classes, may be performed in ArcMap
and then published as a service. Related records may then be displayed
in pop-up windows in ArcGIS Online.
Depending on how the relationship class is created, users are able to
add or update features and have the related data be updated in the
pop-up as well.
However, related tables cannot natively be created in ArcGIS Online,
and the in-memory joins or relates are not supported in a feature
Temporary joins are not visible in ArcGIS Online. Joins from Join
Fields persist in ArcGIS Online as the fields are permanently appended
to the table.
It appears that you cannot produce a join or relate in AGOL using your web-hosted spreadsheet. You would have to add the google table information to ArcMap and create a relate OR create fields in your hosted layer for the join and then attempt to join in the appropriate data from the table. I would then attempt to markup a join by editing the service definition draft or SSD. I have never done this before, but I am used to editing the SDD and I will warn you that it involves a lot of risk so practice on a COPY.
Alternatively, you can add your data to your google spreadsheet since the feature service data is static, and the spreadsheet data is dynamic, and just visualize the published spreadsheet standalone and you don't even have to worry about a join.
If you don't need to JOIN and just need to ADD the data from google, you can follow this blog post: https://blogs.esri.com/esri/arcgis/2012/02/21/using-google-docs-in-your-arcgis-online-maps/
But I think you already know this... Alternatively, google has it's own mapping app for mapping spreadsheets: http://www.google.com/earth/outreach/tutorials/spreadsheet3.html
Lastly, you can cut google spreadsheets out of the operation entirely, and just use AGOL since they allow you to manage and edit data as well as track edits.