Hello, Jerry.
I might have a solution for you. Note that this is a workaround, but it does allow Outlook and GQ's to be much more integrated than what it is out of the box.
Step 1: Check your GQ Settings
First thing we need to do is grab your GQ's Inbox email and make sure the email settings are correct.
Copy/paste your inbox address (you'll need that in a moment) and make sure the
Add sender's email address.. is clicked. You'll use this to help reference information. I would also suggest you make certain
Enable Quick Add Syntax on incoming emails not be selected, as strange characters in email subject lines can cause problems. After completing this step, save and move to the next.

Step 2: Create a New Folder in Outlook
Create a new folder in Outlook that will hold a copy of your emails that are directly related or reference work to be done. I titled mine "GQ To Do" and place it directly under my inbox. Once you have completed this step, move to the next.
Step 3: Create A Quick StepQuick steps are great for completing several different actions with a single click of the button. We will create a quick step titled "To GQ To Do" that will take the highlighted email and move it over to your "GQ To Do" folder and send it to GQ's.

Suggested additional option values (click
Show Options):
Step 4: Use It
Here's how it works. You get an email that requests an action or requires an action. Highlight it and click the "GQ To DO" Quick Step. This will do 2 things. First it will open a new email with the selected email as an inline forward. The address is already set, as is the body and subject. Type in any other additional notes you like and click send.
When you send the email, the forwarded email will be sent to the "GQ To Do" folder and a detailed email will be sent to GQ's inbox. If you need to reference the original email, the original information is there, along with the name and date, making it easy to find in your "GQ To Do" folder. Any attachments will be sent to Google Drive.
Step 5: Clear UpWithout true integration, you will need to do clean your "GQ To Do" folder manually. Completing a task in GQ's does not notify you that it might have an associated email. You'll need to make it part of your process to check the task's notes to determine if their is an associated email in your Outlook. This is really nothing different than what we do with Google Drive. Just something to be aware of.
And that's it! I hope it works for you. It has worked awesome for me.