To convert the given natural language description into a symbolic representation and then into Gurobi code, we first identify the variables and their corresponding natural language objects. We have two main variables: 'hours worked by Hank' and 'hours worked by Jean'. Let's denote these as `x1` and `x2`, respectively.

The objective function is to minimize \(3 \times (x1)^2 + 1 \times x1 \times x2 + 1 \times x1\).

Given the constraints, we can represent them symbolically. For instance, the total combined dollar cost per hour from hours worked by Hank and Jean must be greater than or equal to 61, which translates to \(1x1 + 6x2 \geq 61\), considering Hank's dollar cost per hour is 1 and Jean's is 6.

Here's a symbolic representation of the problem:

```json
{
    'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Jean')],
    'objective_function': '3*x1**2 + x1*x2 + x1',
    'constraints': [
        'x1 >= 0', 'x2 >= 0',  # Non-negativity constraints
        '17*x1 + 11*x2 >= 52',  # Combined computer competence rating
        '12*x1 + 7*x2 >= 69',   # Combined likelihood to quit index
        '1*x1 + 14*x2 >= 25',   # Combined organization score
        '3*x1**2 - 10*x2**2 >= 0',  # Specific quadratic constraint
        'x1 + 6*x2 <= 133',      # Upper bound on combined dollar cost per hour
        '(17*x1)**2 + (11*x2)**2 <= 108',  # Upper bound on combined squared computer competence rating
        '12*x1 + 7*x2 <= 165',   # Upper bound on combined likelihood to quit index
        '(1*x1)**2 + (14*x2)**2 <= 35'    # Upper bound on combined squared organization score
    ]
}
```

To implement this problem in Gurobi, we'll use Python. Note that the constraints and objective function are directly translated from the symbolic representation.

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(name='hours_worked_by_Hank', lb=0)
x2 = m.addVar(name='hours_worked_by_Jean', lb=0)

# Objective function
m.setObjective(3*x1**2 + x1*x2 + x1, GRB.MINIMIZE)

# Constraints
m.addConstr(17*x1 + 11*x2 >= 52, name='combined_computer_competence')
m.addConstr(12*x1 + 7*x2 >= 69, name='combined_likelihood_to_quit')
m.addConstr(x1 + 14*x2 >= 25, name='combined_organization_score')
m.addConstr(3*x1**2 - 10*x2**2 >= 0, name='quadratic_constraint')
m.addConstr(x1 + 6*x2 <= 133, name='upper_bound_dollar_cost')
m.addConstr((17*x1)**2 + (11*x2)**2 <= 108, name='upper_bound_squared_computer_competence')
m.addConstr(12*x1 + 7*x2 <= 165, name='upper_bound_likelihood_to_quit')
m.addConstr((x1)**2 + (14*x2)**2 <= 35, name='upper_bound_squared_organization_score')

# Optimize
m.optimize()

if m.status == GRB.OPTIMAL:
    print('Optimal solution found:')
    print(f'Hours worked by Hank: {x1.X}')
    print(f'Hours worked by Jean: {x2.X}')
else:
    print('No optimal solution found')
```