Unity noob needs help rotating with Quaternions

It sounds like you're going to want to use euler angles instead of quaternions in this case.

The script example you posted in the comments probably isn't working because you're checking GetButtonDown() instead of GetButton(), which means it's just going to check the first frame a button was pressed down. Changing it to GetButton() will call the rotate method constantly, as desired.

However, while Transform.Rotate() is an acceptable solution, I think you would do better to have direct control of the rotation at every point, and it wouldn't take much more coding.

By keeping a variable to hold the transform's local euler angles, we can always quickly check or adjust them to suit a desire. With Rotate, we're telling it to perform an action, and storing no information about the result.

Here's one way to use a stored rotation value to control your flipper:

 using UnityEngine;
 using System.Collections;
 
 public class FlipperScript : MonoBehaviour {
 
     public Transform flipperLeftTop;

     public float FlipSpeed; // controls how fast the flipper moves when button is pressed
     public float ResetSpeed;// controls how fast it returns to rest position
     
     private float maxZ;
     private Vector3 flipperRotation = Vector3.zero; 

     // Use this for initialization
     void Start () {
         maxZ = 45f;
     }
     
     // Update is called once per frame
     void Update () {
         if (Input.GetButton("Left Flipper") )
         {
         
             // update the stored z rotation value, limited to the max z, at an adjustable speed
             flipperRotation.z += Time.deltaTime * FlipSpeed;
             if (flipperRotation.z > maxZ) flipperRotation.z = maxZ;
         }
         else
         { 
                     // if the button isn't pressed, rotate it toward the rest position
             flipperRotation.z -= Time.deltaTime * ResetSpeed;
             if (flipperRotation.z < 0f) flipperRotation.z = 0f;
         }
         
         flipperLeftTop.localEulerAngles = flipperRotation; // set the transform to match our stored rotation.
     }
 }

I changed flipperLeftTop from GameObject to Transform, because you were only using the transform of it and not the game object, in the sample provided. This saves keystrokes and improves performance (since you're not fetching the transform every frame, it's already fetched) slightly.

The idea is to keep a local representation of the local euler angles of the object. The euler angles are the more familiar terms of rotation you'd be looking for. Setting local euler's z value to 20f, for example, will equate to 20 degrees of rotation around the z axis. Setting a quaternion value will not have the same effect.

By keeping the flipperRotation a local variable, you have complete control over it, and can assign it to localEulerAngles when ready. Before testing, make sure to set your FlipSpeed and ResetSpeed variables in the inspector, as leaving them at 0f will prevent the flipper from moving.