I am creating a secure web based API that uses HTTPS; however, if I allow the users to configure it (include sending password) using a query string will this also be secure
From a "sniff the network packet" point of view a GET request is safe, as the browser will first establish the secure connection and then send the request containing the GET parameters. But GET url's will be stored in the users browser history / autocomplete, which is not a good place to store e.g. password data in. Of course this only applies if you take the broader "Webservice" definition that might access the service from a browser, if you access it only from your custom application this should not be a problem.
So using post at least for password dialogs should be preferred. Also as pointed out in the link littlegeek posted a GET URL is more likely to be written to your server logs.