Sure! Here is a test case generator for the given problem statement:

```python
import random

def generate_test_case():
    # Generate random values for H and W within the constraints
    H = random.randint(1, 500)
    W = random.randint(1, 500)
    
    # Generate random values for the grid a_ij within the constraints
    grid = [[random.randint(0, 9) for _ in range(W)] for _ in range(H)]
    
    # Convert the grid to a string format for easy display
    grid_str = "\n".join(" ".join(map(str, row)) for row in grid)
    
    return H, W, grid_str

# Example usage
if __name__ == "__main__":
    H, W, grid_str = generate_test_case()
    print(f"{H} {W}")
    print(grid_str)
```

This function generates random test cases for the grid problem by:
1. Randomly selecting the dimensions of the grid `H` and `W` within the given constraints.
2. Filling the grid with random values between 0 and 9 inclusive.
3. Converting the grid to a string format to be printed in the required format.

You can call `generate_test_case` to get a new test case each time. Here is an example of how the output might look:

```
3 4
4 0 7 3
9 6 2 8
1 0 0 5
```

This is a randomly generated test case with a 3x4 grid. Each call to `generate_test_case` will yield different values for the grid.