To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables for each object in the problem and then representing the objective function and constraints using these variables.

Let's denote:
- $x_0$ as the hours worked by George,
- $x_1$ as the hours worked by Bill.

The objective function to maximize is given as:
\[7.22x_0^2 + 3.77x_0x_1 + 7.44x_0 + 1.6x_1\]

The constraints are as follows:
1. Organization score constraint: $16x_0 + x_1 \geq 37$
2. Productivity rating constraint: $14x_0 + 17x_1 \geq 25$
3. Quadratic constraint: $-x_0^2 + x_1^2 \geq 0$
4. Combined organization score from squared hours worked: $16x_0^2 + x_1^2 \leq 79$
5. Combined organization score from hours worked: $16x_0 + x_1 \leq 79$
6. Combined productivity rating constraint: $14x_0 + 17x_1 \leq 81$

Given the fractional and non-restricted nature of hours worked by both George and Bill, we consider both variables as continuous.

The symbolic representation in JSON format is:

```json
{
    'sym_variables': [('x0', 'hours worked by George'), ('x1', 'hours worked by Bill')],
    'objective_function': '7.22*x0**2 + 3.77*x0*x1 + 7.44*x0 + 1.6*x1',
    'constraints': [
        '16*x0 + x1 >= 37',
        '14*x0 + 17*x1 >= 25',
        '-x0**2 + x1**2 >= 0',
        '16*x0**2 + x1**2 <= 79',
        '16*x0 + x1 <= 79',
        '14*x0 + 17*x1 <= 81'
    ]
}
```

To solve this problem using Gurobi, we'll write the Python code as follows:

```python
from gurobipy import *

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

# Define variables
x0 = m.addVar(name='hours_worked_by_George', lb=-GRB.INFINITY, ub=GRB.INFINITY)
x1 = m.addVar(name='hours_worked_by_Bill', lb=-GRB.INFINITY, ub=GRB.INFINITY)

# Objective function: Maximize
m.setObjective(7.22*x0**2 + 3.77*x0*x1 + 7.44*x0 + 1.6*x1, GRB.MAXIMIZE)

# Constraints
m.addConstr(16*x0 + x1 >= 37, name='organization_score_min')
m.addConstr(14*x0 + 17*x1 >= 25, name='productivity_rating_min')
m.addConstr(-x0**2 + x1**2 >= 0, name='quadratic_constraint')
m.addConstr(16*x0**2 + x1**2 <= 79, name='combined_organization_score_max_squared')
m.addConstr(16*x0 + x1 <= 79, name='combined_organization_score_max')
m.addConstr(14*x0 + 17*x1 <= 81, name='combined_productivity_rating_max')

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by George: {x0.x}")
    print(f"Hours worked by Bill: {x1.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```