To find the solution, let's break down the problem into smaller steps:

1. Given Equation:
We are given the equation 2a + 3b + 6c = 0. This equation represents a linear equation in three variables (a, b, c).

2. Quadratic Equation:
The equation we need to analyze is ax^2 + bx + c = 0. This is a quadratic equation with the variables a, b, and c.

3. Relationship between Linear and Quadratic Equations:
For a quadratic equation ax^2 + bx + c = 0, the coefficients a, b, and c are related to the roots of the equation. The sum of the roots is -b/a, and the product of the roots is c/a.

4. Applying the Relationship:
Using the given equation 2a + 3b + 6c = 0, we can observe that the sum of the coefficients is 2 + 3 + 6 = 11. Therefore, the sum of the roots of the quadratic equation is -b/a = -11/a.

5. Analyzing the Possible Roots:
To determine if there is at least one root in the interval (0, 1), we need to analyze the values of -b/a within this interval.

Let's consider the extreme cases:
- If a is positive, then for -b/a to be within the interval (0, 1), b must be negative.
- If a is negative, then for -b/a to be within the interval (0, 1), b must be positive.

6. Conclusion:
Since in the given equation 2a + 3b + 6c = 0, the coefficient of b is positive (3), we can conclude that a must also be positive for -b/a to be within the interval (0, 1). Therefore, at least one root of the quadratic equation ax^2 + bx + c = 0 lies in the interval (0, 1).

7. Final Answer:
Hence, the correct answer is option 'D' (0, 1).