Is there anyway of adding a custom query string parameter after clicking the Save & Close button?

You can include any query parameter you want in the URL so long as you do not conflict with  existing parameters - QuickBase will just ignore it. But you will have to arrange for script run in the page using an injection technique -  such as IOL -  that will read the URL parameter and synthesize a click on the appropriate tab. The relevant code to extract the query string parameter is this:

var urlParams = new URLSearchParams(window.location.search);
var customValue  = urlParams.get("customeQueryStringParam");