onRequestPermissionsResult() method is deprecated in androidx.fragment.app.Fragment.

So you use registerForActivityResult() method instead onRequestPermissionsResult().

You can refer this URL.

Following is kotlin code. but you can refer it.

val permReqLuncher = registerForActivityResult(ActivityResultContracts.RequestPermission()){
  if (it) {
     // Good pass
  } else {
     // Failed pass
  }
}

I added java code from following URL.
How to get a permission request in new ActivityResult API (1.3.0-alpha05)?

private ActivityResultLauncher<String> mPermissionResult = registerForActivityResult(
        new ActivityResultContracts.RequestPermission(),
        new ActivityResultCallback<Boolean>() {
            @Override
            public void onActivityResult(Boolean result) {
                if(result) {
                    Log.e(TAG, "onActivityResult: PERMISSION GRANTED");
                } else {
                    Log.e(TAG, "onActivityResult: PERMISSION DENIED");
                }
            }
        });



        // Launch the permission window -- this is in onCreateView()
    floatingActionButton.setOnClickListener(new View.OnClickListener() {
        @Override
        public void onClick(View v) {
         mPermissionResult.launch(Manifest.permission.ACCESS_BACKGROUND_LOCATION);

        }
    });

You can request multiple permissions.

    val requestMultiplePermissions = registerForActivityResult(ActivityResultContracts.RequestMultiplePermissions()) { permissions ->
        permissions.entries.forEach {
            Log.e("DEBUG", "${it.key} = ${it.value}")
        }
    }

    requestMultiplePermissions.launch(
        arrayOf(
            Manifest.permission.READ_CONTACTS,
            Manifest.permission.ACCESS_FINE_LOCATION
       )
    )

You can use some external library for permission handling to reduce some boilerplate code. I use Dexter library. Rx Permissions is also good choice if you are using RxJava2.

This works for me - (kotlin):

private fun checkPermissions() {
    if (mContext?.let {
        ContextCompat.checkSelfPermission(
            it,
            READ_EXTERNAL_STORAGE
        )
    } != PackageManager.PERMISSION_GRANTED) {
    Log.d(TAG, "Request Permissions")
    requestMultiplePermissions.launch(
        arrayOf(READ_EXTERNAL_STORAGE, WRITE_EXTERNAL_STORAGE))
 } else {
        Log.d(TAG, "Permission Already Granted")
    }
}

private val requestMultiplePermissions =
    registerForActivityResult(ActivityResultContracts.RequestMultiplePermissions()) { permissions ->
        permissions.entries.forEach {
            Log.d(TAG, "${it.key} = ${it.value}")
        }
        if (permissions[READ_EXTERNAL_STORAGE] == true && permissions[WRITE_EXTERNAL_STORAGE] == true) {
            Log.d(TAG, "Permission granted")
        } else {
            Log.d(TAG, "Permission not granted")
        }
    }

A simple way in Kotlin

import android.Manifest
import android.content.Context
import android.content.pm.PackageManager
import android.os.Build
import androidx.activity.result.contract.ActivityResultContracts
import androidx.core.app.ActivityCompat
import androidx.fragment.app.Fragment

class MyFragment : Fragment() {

companion object {
    val TAG: String = MyFragment::class.java.simpleName
    var PERMISSIONS = arrayOf(
        Manifest.permission.CAMERA,
        Manifest.permission.READ_EXTERNAL_STORAGE,
        Manifest.permission.WRITE_EXTERNAL_STORAGE
    )
}

private val permReqLauncher =
    registerForActivityResult(ActivityResultContracts.RequestMultiplePermissions()) { permissions ->
        val granted = permissions.entries.all {
            it.value == true
        }
        if (granted) {
            displayCameraFragment()
        }
    }

private fun takePhoto() {
    if (Build.VERSION.SDK_INT < Build.VERSION_CODES.M) {
        displayCameraFragment()
    }
    activity?.let {
        if (hasPermissions(activity as Context, PERMISSIONS)) {
            displayCameraFragment()
        } else {
            permReqLauncher.launch(
                PERMISSIONS
            )
        }
    }
}

// util method
private fun hasPermissions(context: Context, permissions: Array<String>): Boolean = permissions.all {
    ActivityCompat.checkSelfPermission(context, it) == PackageManager.PERMISSION_GRANTED
}

private fun displayCameraFragment() {
    // open camera fragment
}
}

Alternative registerForActivityResult

Example use:

 private fun permissionSetup() {
    val permission = ContextCompat.checkSelfPermission(
        requireContext(), Manifest.permission.READ_CONTACTS)

    if (permission != PackageManager.PERMISSION_GRANTED) {
        permissionsResultCallback.launch(Manifest.permission.READ_CONTACTS)
    } else {
        println("Permission isGranted")
    }
}

private val permissionsResultCallback = registerForActivityResult(
    ActivityResultContracts.RequestPermission()){
    when (it) {
        true -> { println("Permission has been granted by user") }
        false -> {
            Toast.makeText(requireContext(), "Permission denied", Toast.LENGTH_SHORT).show()
            dialog.dismiss()
        }
    }
}

Most of the answers address the OP requirement. But I have found few things that are missing so I thought to provide a complete example (in Koltin)

class ProfileFragment : Fragment(){

private lateinit var permissionRequest : ActivityResultLauncher<Array<String>>

companion object {
      
    val LOCATION_PERMISSIONS = arrayOf(
            Manifest.permission.ACCESS_FINE_LOCATION,
            Manifest.permission.ACCESS_COARSE_LOCATION
        )
    }

    private fun getGpsLocation() {
        if(activity != null){
            permissionRequest.launch(LOCATION_PERMISSIONS)
        }
    }

    override fun onViewCreated(view: View, savedInstanceState: Bundle?) {
        super.onViewCreated(view, savedInstanceState)

        binding.locBtn.setOnClickListener { getGpsLocation() }
    }


    override fun onCreate(savedInstanceState: Bundle?) {
        super.onCreate(savedInstanceState)
        registerPermissionRequest()
    }
        
    override fun onCreateView(inflater: LayoutInflater, container: ViewGroup?, savedInstanceState: Bundle?): View {

        binding = DataBindingUtil.inflate(inflater, R.layout.fragment_profile, container, false)
       
        return binding.root
    }
    
    private fun registerPermissionRequest(){
            var permissionCount = 0
            permissionRequest = registerForActivityResult(ActivityResultContracts.RequestMultiplePermissions()) { permissions ->
                permissions.entries.forEach {
                    if(it.value){
                        permissionCount++
                    }
    
                }
    
                if(permissionCount == 2){
                    getMyLocation()
                }
            }
        }
    }

The things that are missing are

A)Fragments must call registerForActivityResult() before they are created (i.e. initialization, onAttach(), or onCreate()). Other wise it won't work and the app would crash.

Error:

java.lang.IllegalStateException: Fragment ProfileFragment{bf12414} (210ad5a1-3286-4586-a48f-deac1d8e3eef id=0x7f09008b) is attempting to registerForActivityResult after being created. Fragments must call registerForActivityResult() before they are created (i.e. initialization, onAttach(), or onCreate()).

B) It is recommended to request permission when it is really needed. In my example when user clicks on Button with id locBtn, permission dialog is shown rather than showing when activity/fragment is created.

Please refer to the official documentation: https://developer.android.com/training/permissions/requesting

Review the following, which can be found in the documentation.

// Register the permissions callback, which handles the user's response to the
// system permissions dialog. Save the return value, an instance of
// ActivityResultLauncher. You can use either a val, as shown in this snippet,
// or a lateinit var in your onAttach() or onCreate() method.
val requestPermissionLauncher =
    registerForActivityResult(RequestPermission()
    ) { isGranted: Boolean ->
        if (isGranted) {
            // Permission is granted. Continue the action or workflow in your
            // app.
        } else {
            // Explain to the user that the feature is unavailable because the
            // features requires a permission that the user has denied. At the
            // same time, respect the user's decision. Don't link to system
            // settings in an effort to convince the user to change their
            // decision.
        }
    }

After which, launch the request

when {
    ContextCompat.checkSelfPermission(
            CONTEXT,
            Manifest.permission.REQUESTED_PERMISSION
            ) == PackageManager.PERMISSION_GRANTED -> {
        // You can use the API that requires the permission.
    }
    //Is not needed for it to work, but is a good practice as it plays a role
    //in letting user know why the permission is needed.
    shouldShowRequestPermissionRationale(...) -> {
        // In an educational UI, explain to the user why your app requires this
        // permission for a specific feature to behave as expected. In this UI,
        // include a "cancel" or "no thanks" button that allows the user to
        // continue using your app without granting the permission.
        showInContextUI(...)
    }
    else -> {
        // You can directly ask for the permission.
        // The registered ActivityResultCallback gets the result of this request.
        requestPermissionLauncher.launch(
                Manifest.permission.REQUESTED_PERMISSION)
    }
}