Sure! Here is a Python function to generate test cases for the "DAG Trio" problem:

```python
import random

def generate_dag_trio_test_case():
    N = random.randint(3, 500)
    M = random.randint(max(3, N-1), 30000)
    
    edges = set()
    while len(edges) < M:
        a = random.randint(1, N)
        b = random.randint(1, N)
        if a != b and (a, b) not in edges and (b, a) not in edges:
            edges.add((a, b))
    
    edges = list(edges)
    random.shuffle(edges)
    
    test_case = f"{N} {M}\n"
    for a, b in edges:
        test_case += f"{a} {b}\n"
    
    return test_case.strip()

# Example usage:
print(generate_dag_trio_test_case())
```

### Explanation:

1. **Vertices and Edges Count:**
    - `N` is chosen randomly between 3 and 500.
    - `M` is chosen randomly between the maximum of 3 and \(N-1\) to 30000.

2. **Edges Generation:**
    - A set `edges` is used to store unique edges.
    - Random edges `(a, b)` are generated ensuring:
        - `a` is not equal to `b` (no self-loops).
        - Each edge `(a, b)` is unique and not already in the set (no multiple edges).
        - Ensuring the graph is simple and connected (in practice, this random method combined with the constraints helps generate a connected graph).

3. **Output Format:**
    - The test case is returned as a formatted string where:
        - The first line is `N M`.
        - Each subsequent line represents an edge `a b`.

This generator will help create randomized test cases suitable for the "DAG Trio" problem while respecting the constraints provided.