When migrating from MySite on-premise to OneDrive in the cloud, Quick Links are not included by third party utilities (i.e. ShareGate).  However, with PowerShell we can export the original raw CSV data and provide a list of links to end users.  They can bookmark or add to Office 365.

Cheers! 

Code

# User login parameter
$userLogin = "johndoe"
$mySiteWebApp = Get-SPWebApplication |? {$_.Name -like "*MySite*"}
# Connect to MySite
$site = Get-SPSite $mySiteWebApp.Url
$context = Get-SPServiceContext $site
$profileManager = New-Object Microsoft.Office.Server.UserProfiles.UserProfileManager($context);
# Lookup user profile in UPS
$userProfile = $profileManager.GetUserProfile($env:USERDNSDOMAIN + $userLogin);
# Display on console
$userProfile
$userProfile.QuickLinks.GetItems() | Select Title, Url
# Export to CSV file
$userProfile.QuickLinks.GetItems() | Select Title, Url | Export-Csv "C:\temp\$($userLogin)-FollowedLinks.csv" –NoTypeInformation -Force

Screenshot