If you are using similar quick links web parts on multiple SharePoint site pages and you want to add new links in one place which should be shown in all quick links web parts, try below solutions.
Using JSON formatting:
You can use the SharePoint list to store information related to links (link title, link/URL, etc.) and use JSON formatting to customize the display of links.
Then add "List" web part on home page (where you want to show the links) and show the JSON formatted customized list on the page.
This gives you better flexibility over the design and you can manage (add/remove/hide based on flag column or by creating different filtered views) the quick links from SharePoint list without editing the site pages.
Check below samples which should help you with this:
Icon Link Tiles
Using SharePoint framework (SPFx):
You can also use custom web parts developed using SPFx for showing the quick links dynamically on site pages.
Here are few samples to show the capabilities of SPFx development for quick links:
- Custom Links
- Links Web Part