Hi,
Thank you for sharing the experience with us.
I went through your post carefully. It seems like you set up a Quick links web part on the home page of SharePoint site which is connected to a team in Teams. The Quick links web part links 2 notebooks created in the team. The issue is that the link doesn't work when you click it for the second time.
To narrow down the issue and for troubleshooting, could you please confirm information below?
1.May I know how do you set up notebooks link in the Quick links web part?
By firstly going to the Files tab in teams, find the notebook, select it and click "Copy link", copy the link from SharePoint tab. Then go to the site home page, edit the page and add a Quick links web part, click Add link, choose "From a link" and enter the copied link?
When I use these steps to test from my side, it works fine when I go to click on the link for the second time.
If you use different steps, please let me know.
2.Does same issue happen when you click on the notebooks link from different browsers? (Edge, Chrome, Firefox)
3.Does the issue only happen to the specific 2 notebooks? What about if you try other notebooks from Files tab?
4.If you try a different team and set up links to notebooks on the related site's home page, does issue still happen?
5.You mentioned "The notebooks are stored in the teams file tab", are the notebooks created by Files tab>click New>choose OneNote notebook?
Thanks for your patience and effort. Have a nice day and stay safe!
Best Regards,
Tina