Migrate email to Office 365: https://support.office.com/en-us/article/-a3e3bddb-582e-4133-8670-e61b9f58627e. This is a general landing page that lists all the ways to migrate email to Office 365.
Migrate Gmail mailboxes to Office 365: https://support.office.com/en-us/article/Migrate-Gmail-mailboxes-to-Office-365-665dc56c-581c-4e35-8028-6bc1e8497016
I would like to share the steps to migrate a mailbox from Gmail or any other IMAP provider to Office 365 exchange online. Following steps may be helpful for reference.
Create a CSV file as per the screenshot mentioned below. You should include office 365 email address ,IMAP user account and password in the CSV file.
Login to office 365 portal and select exchange admin center as seen below.
Under recipients , select migration tab.
Select the add button and create a new migration batch.
Select migration to exchange online option from the drop down list.
Choose IMAP migration from the available option and click next.
Browse and select the CSV file where in mailbox information is stored and click next.
Put down the IMAP server FQDN. Gmail's default IMAP server is imap.gmail.com and click next.
Create a batch file name and click next.
You can mention admin's email address to notify the migration status and click next.
You should be able to see the migration batch status as 'syncing'. It will take few minutes based on the mailbox size.
Once migration is completed 'synced' status will change.
Admin will receive an email stating that migration is completed or failed based on batch status.
This step by step guide may help you for IMAP migration.