To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints.

Let's define:
- $x_1$ as 'hours worked by Hank'
- $x_2$ as 'hours worked by Mary'

The objective function is given as: 
\[8.62x_1^2 + 7.29x_1x_2 + 1.66x_2^2 + 9.21x_1 + 9.73x_2\]

Constraints:
1. Paperwork competence rating for Hank: $3x_1$
2. Work quality rating for Hank: $6x_1$
3. Computer competence rating for Hank: $13x_1$
4. Paperwork competence rating for Mary: $13x_2$
5. Work quality rating for Mary: $3x_2$
6. Computer competence rating for Mary: $13x_2$
7. Total combined paperwork competence rating: $3x_1 + 13x_2 \geq 13$
8. Total combined work quality rating: $6x_1 + 3x_2 \geq 22$
9. Total combined computer competence rating: $13x_1 + 13x_2 \geq 16$
10. Linear constraint: $-9x_1 + 4x_2 \geq 0$
11. Upper bound on total paperwork competence: $3x_1 + 13x_2 \leq 19$
12. Upper bound on total work quality rating: $6x_1 + 3x_2 \leq 63$
13. Upper bound on combined computer competence squared: $(13x_1)^2 + (13x_2)^2 \leq 34$ (Note: This constraint seems unusual as it involves squares of linear terms, which is not typical in linear programming. However, we'll translate it directly from the description.)
14. $x_1$ can be non-integer.
15. $x_2$ must be an integer.

Symbolic representation:
```json
{
  'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Mary')],
  'objective_function': '8.62*x1**2 + 7.29*x1*x2 + 1.66*x2**2 + 9.21*x1 + 9.73*x2',
  'constraints': [
    '3*x1 + 13*x2 >= 13', 
    '6*x1 + 3*x2 >= 22', 
    '13*x1 + 13*x2 >= 16', 
    '-9*x1 + 4*x2 >= 0', 
    '3*x1 + 13*x2 <= 19', 
    '6*x1 + 3*x2 <= 63', 
    '(13*x1)**2 + (13*x2)**2 <= 34'
  ]
}
```

Now, translating this into Gurobi code:

```python
from gurobipy import *

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

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Hank")
x2 = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Mary")

# Set the objective function
m.setObjective(8.62*x1**2 + 7.29*x1*x2 + 1.66*x2**2 + 9.21*x1 + 9.73*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(3*x1 + 13*x2 >= 13, name="paperwork_competence")
m.addConstr(6*x1 + 3*x2 >= 22, name="work_quality_rating")
m.addConstr(13*x1 + 13*x2 >= 16, name="computer_competence")
m.addConstr(-9*x1 + 4*x2 >= 0, name="linear_constraint")
m.addConstr(3*x1 + 13*x2 <= 19, name="upper_bound_paperwork")
m.addConstr(6*x1 + 3*x2 <= 63, name="upper_bound_work_quality")
m.addConstr((13*x1)**2 + (13*x2)**2 <= 34, name="upper_bound_computer_squared")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print("Hours worked by Hank:", x1.x)
    print("Hours worked by Mary:", x2.x)
else:
    print("No optimal solution found.")
```