What Fallacy is this?

You do have reason to believe I have had miracles. Because you are not an idiot who is incapable of judging truth and lie.

I'm discussing miracles with someone and this is a statement made by the other person. I think it's a False Dichotomy and there is no good way to answer this. I have to agree or else I'm an idiot who is incapable of judging truth or lies. This seems incredibly unfair and there are other possibilities.

Is this a False Dichotomy ?

Yes, I would say that's a false dilemma<>. If you check the description of the false dilemma fallacy on this site, you'll find the following:

False dilemmas are usually characterized by “either this or that” language, but can also be characterized by omissions of choices.

The example provided for the omission of choices false dilemma is:

Example (omission):
I thought you were a good person, but you weren’t at church today.
Explanation: The assumption here is that if one doesn't attend chuch, one must be bad. Of course, good people exist who don’t go to church, and good church-going people could have had a really good reason not to be in church -- like a hangover from the swingers' gathering the night before.

This is similar to what that person has said to you regarding miracles:

You do have reason to believe I have had miracles. Because you are not an idiot who is incapable of judging truth and lie.

So you either believe his personal account of having experienced miracles, or you are an idiot. Of course, a lot of other options exist, like you simply not being convinced that his reasons are adequate to conclude miracles, or you can diagnose the person with hallucinations, etc.

Yes. It is an implied false dichotomy . In addition, an implied ad hominem and perhaps even a sprinkle of argument by intimidation .

I believe Miracles happen because I've experienced them.
You're not an idiot so you know I speak the truth
Therefore you also have reason to believe in miracles

Hmmmm, I smell a bit of Ad hoc rescue with a
smidgen of Appeal to Flattery:

Appeal to Flattery
(also known as: apple polishing, wheel greasing, brown nosing, appeal to pride / argumentum ad superbiam, appeal to vanity)

Description: When an attempt is made to win support for an argument, not by the strength of the argument, but by using flattery on those whom you want to accept your argument. This fallacy is often the cause of people getting tricked into doing something they don’t really want to do.

Logical Form:

X is true.
(flattery goes here)
Therefore, X is true.
Example #1:

You should certainly be the one who washes the dishes -- you are just so good at it!
Explanation: You may be great at washing dishes, but that fact in itself is not a sufficient reason for you being the one to wash the dishes. Is it necessary for someone as skilled at dish-washing as you to do the job, or is it a mindless job that anyone can do just fine?

Example #2:

Salesguy: You should definitely buy this car. You look so good in it -- you look at least ten years younger behind that wheel.
Tamera: I’ll take it!
Explanation: The comment about looking ten years younger just because of the car is obvious flattery and not a fact. This would not qualify as a valid reason for making such a purchase.

Exception: Sincere praise is not flattery and is universally appreciated[1]. However, even praise in itself, without reason, is a fallacy, unless the argument is directly related to the sincere praise.

You are a stunningly beautiful girl -- you should be a model.
Tip: Flattery might get you somewhere, but it’s usually a place you don’t want to be.