No way to achieve this just using Hubspot. One thing you can do is send the date to zapier, run an operation on Zapier to extract the quartely date and then extract the date back to from Zapier and pass it to the new field in Hubspot.
We shouldn't have to use an external third party application to carry out basic date functionality in HubSpot. However, I will go and search web for Zapier again. Come on HubSpot, please be better than this.
To give some context on this, when the 'Last QBR date' is populated, I would like for the highlighted property to be auto filled for 3 months from that date.
No way to achieve this just using Hubspot. One thing you can do is send the date to zapier, run an operation on Zapier to extract the quartely date and then extract the date back to from Zapier and pass it to the new field in Hubspot.