To solve the given optimization problem using Gurobi, we need to formulate it as a mathematical program. The objective function and constraints provided in the natural language description will be translated into a Python code that utilizes the Gurobi library.

First, let's clarify the variables and their constraints:
- Let \(L\), \(P\), and \(B\) represent the hours worked by Laura, Peggy, and Bill, respectively.
- The objective function to maximize is: \(9L^2 + 2LP + 3LB + 9P^2 + 8PB + 9B^2 + 3L + 2P + 9B\).
- Constraints:
  - Paperwork competence ratings: \(6L\), \(3P\), and \(2B\).
  - Likelihood to quit indices: \(7L\), \(6P\), and \(14B\).
  - Minimum total likelihood to quit index from Laura and Bill: \(7L + 14B \geq 20\).
  - Total likelihood to quit index from all three must be at least 30: \(7L + 6P + 14B \geq 30\).
  - Maximum combined paperwork competence rating from Peggy and Bill: \(3P + 2B \leq 92\).
  - Maximum combined paperwork competence rating from Laura and Peggy: \(6L + 3P \leq 100\).
  - Maximum combined paperwork competence rating from all three: \(6L + 3P + 2B \leq 100\).
  - Maximum total likelihood to quit index from Peggy and Bill: \(6P + 14B \leq 121\).
  - Maximum total likelihood to quit index from all three: \(7L + 6P + 14B \leq 76\).

Given these constraints, we notice that there are inconsistencies in the problem statement regarding the maximum total likelihood to quit index from all three workers. It's stated both as a maximum of 76 and without a clear resolution on how to handle this inconsistency, we'll proceed with the most restrictive constraint for maximization problems which typically aim to stay within given bounds.

For simplicity and adherence to instructions, let's focus on translating these into Gurobi code directly:

```python
from gurobipy import *

# Create a model
m = Model("Optimization_Problem")

# Define variables - Hours worked by each person
L = m.addVar(lb=0, name="hours_worked_by_Laura")
P = m.addVar(lb=0, name="hours_worked_by_Peggy")
B = m.addVar(lb=0, vtype=GRB.INTEGER, name="hours_worked_by_Bill")

# Objective function
m.setObjective(9*L**2 + 2*L*P + 3*L*B + 9*P**2 + 8*P*B + 9*B**2 + 3*L + 2*P + 9*B, GRB.MAXIMIZE)

# Constraints
m.addConstr(7*L + 14*B >= 20, name="min_likelihood_to_quit_Laura_Bill")
m.addConstr(7*L + 6*P + 14*B >= 30, name="total_likelihood_to_quit_all")
m.addConstr(3*P + 2*B <= 92, name="max_paperwork_Peggy_Bill")
m.addConstr(6*L + 3*P <= 100, name="max_paperwork_Laura_Peggy")
m.addConstr(6*L + 3*P + 2*B <= 100, name="max_total_paperwork_all")
m.addConstr(6*P + 14*B <= 121, name="max_likelihood_to_quit_Peggy_Bill")
m.addConstr(7*L + 6*P + 14*B <= 76, name="max_total_likelihood_to_quit_all")

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Laura: {L.x}")
    print(f"Hours worked by Peggy: {P.x}")
    print(f"Hours worked by Bill: {B.x}")
else:
    print("No optimal solution found.")
```