I've not got a huge amount of experience with android-query but this looks like this example from their async API documentation should do the trick;
String url = "http://www.mysite.com/Services/GetJson";
AjaxCallback<JSONObject> cb = new AjaxCallback<JSONObject>();
cb.url(url).type(JSONObject.class).weakHandler(this, "jsonCb");
cb.header("Content-Type", "application/json; charset=utf-8");
cb.param("nNumResults", "100");
aq.ajax(cb);
Note that when setting params you can also use a Map;
Map<String,String> params = new HashMap<>();
cb.params(params);
Watchout for the gotcha when working with the param method. It will default the "Content-Type" header for you if you haven't already set it.
You should then get back a response to this.jsonCb(String url, JSONObject jsonObject, AjaxStatus status)
.