Basically you want to get your current URL via JavaScript with:
var existingUrl = window.location.href; // http://localhost/?val=1
Then append any Query Strings that are applicable using:
window.location.href = existingUrl + '&var2=2';
or some other similar code. Take a look at this post about Query Parameters.
Note: A link would already have to exist with an OnClick event that calls a function with the above code in it for it to work appropriately.
Now obviously this isn't very useful information on it's own, so you are going to want to do some work either in JavaScript or in Server code (through use of NodeJS, PHP, or some other server-side language) to pass those variable names and their values down so that the button can do what you are wanting it to do.
You will have to have some logic to make sure the query parameters are put in the URL correctly though. I.E. if there is only one query param it's going to look like '?var1=1' and if it's any subsequent parameter it's going to look like '&var#=#'.