Eskil Abrahamsen Blomfeldt has a post on his blog showing how to call QtActivity/(HangmanActivity) static Java methods from C++ here.
He mentions that he only uses static methods because this is easier (indeed) and nothing more is needed in his example. Quote:
In my game, the Activity is a singleton, so I store a static reference to the object in the constructor. (My C++ Data class has the same logic. I’m doing this so that I can facilitate the communication between the Java and the C++ code using static methods. For a more complex example, it’s also possible to store references and pointers in each C++ and Java object that maps it to its equivalent in the other language, but that is not necessary in this game.)
Calling the static method works. Calling the non-static method does not work(function is never entered). If I call the non-static method connectBluetooth() in onCreate() it works. Is there a way to be able to call and not have the non-static method in onCreate()?
EDIT:
I figured I'll add some code. I have extended the QtActivity class:
package org.qtproject.qt5.android.bindings;
import org.qtproject.qt5.android.bindings.QtApplication;
import org.qtproject.qt5.android.bindings.QtActivity;
import android.bluetooth.BluetoothAdapter;
import android.content.Intent;
import android.os.Bundle;
import android.util.Log;
import java.lang.String;
public class MyActivity extends QtActivity
{
    private BluetoothAdapter bluetoothAdapter;
    private static final int ENABLE_BLUETOOTH_REQUEST = 1;
        public MyActivity()
        {
            Log.d(QtApplication.QtTAG, "MyActivity constructor called");
        }
        @Override
        public void onCreate(Bundle savedInstanceState)
        {
            super.onCreate(savedInstanceState);
        }
        @Override
        public void onDestroy()
        {
            super.onDestroy();
        }
    @Override
        protected void onActivityResult(int requestCode, int resultCode, Intent data)
    {
                Log.d(QtApplication.QtTAG, "onActivityResult entered");
        if(requestCode == 1)
        {
            if(requestCode == 1)
            {
                if(resultCode == RESULT_OK)
                {
                                        Log.d(QtApplication.QtTAG, "User accepted to enable Bluetooth");
                }
                else if(resultCode == RESULT_CANCELED)
                {
                                        Log.d(QtApplication.QtTAG, "User declined to enable Bluetooth");
                }
            }
        }
    }
        public void connectBluetooth()
        {
            Log.d(QtApplication.QtTAG, "connectBluetooth() entered");
            bluetoothAdapter = BluetoothAdapter.getDefaultAdapter();
            if(bluetoothAdapter == null)
            {
                Log.d(QtApplication.QtTAG, "Bluetooth adapter is not found");
                return;
            }
            if(!bluetoothAdapter.isEnabled())
            {
                Log.d(QtApplication.QtTAG, "Bluetooth is off");
                Intent enableBluetoothIntent = new Intent(BluetoothAdapter.ACTION_REQUEST_ENABLE);
                startActivityForResult(enableBluetoothIntent, ENABLE_BLUETOOTH_REQUEST);
            }
        }
        public static void test()
        {
            Log.d(QtApplication.QtTAG, "Static Test OK!");
        }
}
and I try to call these methods from this C++ class:
#include "bluetooth.h"
#include <QtAndroidExtras>
#include <QDebug>
Bluetooth::Bluetooth(QObject *parent) :
    QObject(parent)
{
}
void Bluetooth::connect()
{
    //Test static call
    QAndroidJniObject::callStaticMethod<void>("org/qtproject/qt5/android/bindings/MyActivity", "test");
    //Test non-static call
    //Line below creates a new object and is a subclass of QtActivity.
    bluetooth = new QAndroidJniObject("org/qtproject/qt5/android/bindings/MyActivity");
    if(!bluetooth->isValid())
    {
        qDebug() << "bluetooth is an invalid java object";
        return;
    }
    bluetooth->callMethod<void>("connectBluetooth");
}
EDIT: The part of the program I am most uncertain about is the AndroidManifest.xml. Maybe there is something here that can explain this behaviour?
<?xml version='1.0' encoding='utf-8'?>
<manifest xmlns:android="http://schemas.android.com/apk/res/android" android:versionCode="1" package="org.qtproject.example.AndroidTest" android:versionName="1.0" android:installLocation="auto">
    <application android:label="@string/app_name" android:name="org.qtproject.qt5.android.bindings.QtApplication">
        <activity android:configChanges="orientation|uiMode|screenLayout|screenSize|smallestScreenSize|locale|fontScale|keyboard|keyboardHidden|navigation" android:label="@string/app_name" android:name="org.qtproject.qt5.android.bindings.MyActivity" android:screenOrientation="unspecified">
            <intent-filter>
                <action android:name="android.intent.action.MAIN"/>
                <category android:name="android.intent.category.LAUNCHER"/>
            </intent-filter>
            <meta-data android:value="AndroidTest" android:name="android.app.lib_name"/>
            <meta-data android:resource="@array/qt_sources" android:name="android.app.qt_sources_resource_id"/>
            <meta-data android:value="default" android:name="android.app.repository"/>
            <meta-data android:resource="@array/qt_libs" android:name="android.app.qt_libs_resource_id"/>
            <meta-data android:resource="@array/bundled_libs" android:name="android.app.bundled_libs_resource_id"/>
            <!-- Deploy Qt libs as part of package -->
            <meta-data android:value="1" android:name="android.app.bundle_local_qt_libs"/>
            <meta-data android:resource="@array/bundled_in_lib" android:name="android.app.bundled_in_lib_resource_id"/>
            <meta-data android:resource="@array/bundled_in_assets" android:name="android.app.bundled_in_assets_resource_id"/>
            <!-- Run with local libs -->
            <meta-data android:value="1" android:name="android.app.use_local_qt_libs"/>
            <meta-data android:value="/data/local/tmp/qt/" android:name="android.app.libs_prefix"/>
            <meta-data android:value="plugins/platforms/android/libqtforandroidGL.so:lib/libQt5QuickParticles.so" android:name="android.app.load_local_libs"/>
            <meta-data android:value="jar/QtAndroid.jar:jar/QtAndroidAccessibility.jar:jar/QtAndroid-bundled.jar:jar/QtAndroidAccessibility-bundled.jar" android:name="android.app.load_local_jars"/>
            <meta-data android:value="" android:name="android.app.static_init_classes"/>
            <!--  Messages maps -->
            <meta-data android:value="@string/ministro_not_found_msg" android:name="android.app.ministro_not_found_msg"/>
            <meta-data android:value="@string/ministro_needed_msg" android:name="android.app.ministro_needed_msg"/>
            <meta-data android:value="@string/fatal_error_msg" android:name="android.app.fatal_error_msg"/>
            <!--  Messages maps -->
            <!-- Splash screen -->
            <meta-data android:resource="@layout/splash" android:name="android.app.splash_screen"/>
            <!-- Splash screen -->
        </activity>
    </application>
    <uses-sdk android:minSdkVersion="14" android:targetSdkVersion="19"/>
    <supports-screens android:smallScreens="true" android:anyDensity="true" android:largeScreens="true" android:normalScreens="true"/>
    <uses-permission android:name="android.permission.BLUETOOTH"/>
    <uses-permission android:name="android.permission.BLUETOOTH_ADMIN"/>
    <uses-permission android:name="android.permission.INTERNET"/>
    <uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE"/>
</manifest>
				
                        
I can still not call non-static functions, but a work around is using the singelton pattern which restricts the instantiation of a class to one object and call static functions that are able to call non-static methods.
This is done by adding a class/object reference to itself in the constructor:
It is now possible to call non-static functions in static functions. For example the non-static function bluetoothConnect() above can now be rewritten as:
The only difference here is adding the static keyword to the method and adding "m_instance." before every non-static method call.
This function can now be called from C++ as a static functions:
I guess the downside since this is a singleton is that only one object of this class can be made, but as far as I see it will not limit much..
This was just a workaround, I'm still very interested on how I can call non-static functions. This answer will not be accepted as the accepted answer.