Modifying a query string without reloading the page

If you are looking for Hash modification, your solution works ok. However, if you want to change the query, you can use the pushState, as you said. Here it is an example that might help you to implement it properly. I tested and it worked fine:

if (history.pushState) {
    var newurl = window.location.protocol + "//" + window.location.host + window.location.pathname + '?myNewUrlQuery=1';
    window.history.pushState({path:newurl},'',newurl);
}

It does not reload the page, but it only allows you to change the URL query. You would not be able to change the protocol or the host values. And of course that it requires modern browsers that can process HTML5 History API.

For more information:

http://diveintohtml5.info/history.html

https://developer.mozilla.org/en-US/docs/Web/Guide/API/DOM/Manipulating_the_browser_history

Old question, modern answer to help future devs; using the URL interface:

const url = new URL(window.location);
url.searchParams.set('key', value);
window.history.pushState(null, '', url.toString());

This makes sure you really only change the desired query-parameter.

Building off of Fabio's answer, I created two functions that will probably be useful for anyone stumbling upon this question. With these two functions, you can call insertParam() with a key and value as an argument. It will either add the URL parameter or, if a query param already exists with the same key, it will change that parameter to the new value:

//function to remove query params from a URL
function removeURLParameter(url, parameter) {
    //better to use l.search if you have a location/link object
    var urlparts= url.split('?');   
    if (urlparts.length>=2) {

        var prefix= encodeURIComponent(parameter)+'=';
        var pars= urlparts[1].split(/[&;]/g);

        //reverse iteration as may be destructive
        for (var i= pars.length; i-- > 0;) {    
            //idiom for string.startsWith
            if (pars[i].lastIndexOf(prefix, 0) !== -1) {  
                pars.splice(i, 1);
            }
        }

        url= urlparts[0] + (pars.length > 0 ? '?' + pars.join('&') : "");
        return url;
    } else {
        return url;
    }
}

//function to add/update query params
function insertParam(key, value) {
    if (history.pushState) {
        // var newurl = window.location.protocol + "//" + window.location.host + search.pathname + '?myNewUrlQuery=1';
        var currentUrlWithOutHash = window.location.origin + window.location.pathname + window.location.search;
        var hash = window.location.hash
        //remove any param for the same key
        var currentUrlWithOutHash = removeURLParameter(currentUrlWithOutHash, key);

        //figure out if we need to add the param with a ? or a &
        var queryStart;
        if(currentUrlWithOutHash.indexOf('?') !== -1){
            queryStart = '&';
        } else {
            queryStart = '?';
        }

        var newurl = currentUrlWithOutHash + queryStart + key + '=' + value + hash
        window.history.pushState({path:newurl},'',newurl);
    }
}

I want to improve Fabio's answer and create a function which adds custom key to the URL string without reloading the page.

 function insertUrlParam(key, value) {
        if (history.pushState) {
            let searchParams = new URLSearchParams(window.location.search);
            searchParams.set(key, value);
            let newurl = window.location.protocol + "//" + window.location.host + window.location.pathname + '?' + searchParams.toString();
            window.history.pushState({path: newurl}, '', newurl);
        }
    }

// to remove the specific key
export function removeUrlParameter(paramKey) {
  const url = window.location.href
  console.log("url", url)
  var r = new URL(url)
  r.searchParams.delete(paramKey)
  const newUrl = r.href
  console.log("r.href", newUrl)
  window.history.pushState({ path: newUrl }, '', newUrl)
}