This is because MediaWiki treats this as a parameter with the name [https://www.google.com/search?q
(the value is the rest of the string parameter after =
). You can work around that by using a numbered parameter:
{{missing information|1=[https://www.google.com/search?q=google+search+test This link has an equals sign in it, and the template is not displaying properly.]}}
Things get more complicated if you have more parameters and only some of them are problematic, I'm not sure how does that work. But using numbered parameters for everything should work.