The bonuses stack: they are not coming from the same magical source.
Yes. In general, bonuses stack1 ... J. Crawford, Sage Advice
See the Caveat below for making sure that you discuss this with your DM before the session begins.
To help make your case ...
They are two different magical effects from two different items. There is no text in either description that indicates that they would not stack. What would not stack is two RoP +1.
DMG p. 138
A creature can't attune to more than one copy of an item. For example, a creature can't attune to more than one ring of protection at a time.
The stone of good luck helps with ability checks and saving throws, while the Ring of Protection helps with AC and saving throws. They are not identical items. As long as you have no more than three items attuned, and these two are among them, the rules do not preclude them stacking.
Combining magical effects is covered in the PHB (p. 205). Since each item is different, though they both provide beneficial effects, they should both continue to work.
A Ruling From J. Crawford1
From the Sage Advice compendium. (This is posted on the WoTC site. Unless the AL expressly rejects Sage Advice, it should cover your situation. (Page 16, Sage Advice, Version 1.14, @2016 Wizards of the Coast LLC.)
Do the AC bonuses from a ring of protection and bracers of defense stack?
Yes. In general, bonuses stack, unless they’re from the same spell
(see “Combining Magical Effects” in the Player’s Handbook, p. 205).
You also can’t benefit from more than one ring of protection, for
instance, since you can’t attune to more than one copy of an item at a
Caveat! Discuss it with your DM before the session begins!
Whether or not any given Dungeon Master chooses to utilize Sage Advice as a resource for rules adjudication in D&D Adventurers League play is at the discretion of each individual DM. (A.L. FAQ) (Credit to @Miniman for catching this).