Configuring preferences and options in Outlook.com
Hi SnailShell!
I welcome you to Microsoft Community Support.
I understand that you're having issues connecting your Microsoft account to Gmail.
Don't worry as we're here to help you on your issue.
I strongly suggest that you use Google's two-step verification feature combined with an app password for Outlook:
Google has information on 2-step verification and app passwords here:
https://support.google.com/accounts/answer/180744?hl=en
https://support.google.com/accounts/answer/185833?hl=en&topic=2784804&ctx=topic
While you are at it, using two-step verification for your Microsoft Account is a good idea too!
http://windows.microsoft.com/en-us/windows/two-step-verification-faq
After doing the steps I provided and your issue would still persist, please don't hesitate to contact us back.
Thank you and take care!