Quick links WebPart with Target Audience option

Microsoft is introducing an audience targeting feature for the Quick links web part in SharePoint Online, Check the Roadmap.

Currently it's status is "In development" and it will be available soon for all (January CY2021).

Check more information about this at: Audience Targeting for Quick Links Web Part in SharePoint Online

Enabling audience targeting on the modern page web part is not available by default - there is a SharePoint user voice for this requirement, Microsoft has planned to release this in phases from one type of web part to another type.

The last update from Microsoft was:

"We’re working on this! The first web part we’re enabling it on is the News web part, followed by the highlighted content and events web parts. We’d love to hear about other audience targeting scenarios you’d like us to enable. Thanks!"

Here is the original thread about the above requirement:

Enable Audience targeting on modern page web parts

As an alternate, you can go with developing a custom web part using SPFx framework for the same purpose.