How can I limit my users in the experiment to use Google Chrome and Microsoft Edge browsers only?

You can simply apply a JS to hide next button when user will enter from different browser and give information on same page like “If you want to continue survey and proceed further, please copy the same link and paste it on Chrome or Edge browsers only. Show next button if respondent will enter from Chrome or Edge only.

How can I check which browser it is in?

You can get the information from Qualtrics Meta info variable or from UserAgent metadata embedded field.

Also you can use jQuery to get the information. Try this.

if($.browser.chrome) {
   alert("Google Chrome");
}
if ($.browser.msedge) {
   alert("Microsoft Edge");