To solve the given optimization problem using Gurobi, we first need to define the variables and the objective function. The variables in this case are 'hours worked by Bill' and 'hours worked by John'. The objective is to maximize the function: 5.02 * (hours worked by Bill)^2 + 4.37 * (hours worked by Bill) * (hours worked by John) + 3.47 * (hours worked by John)^2 + 3.78 * (hours worked by John).

The constraints are defined based on the resources/attributes provided:
- Dollar cost per hour for Bill and John
- Work quality rating for Bill and John
- Likelihood to quit index for Bill and John
- Minimum total combined dollar cost per hour from hours worked by both
- Minimum total combined work quality rating
- Minimum total combined likelihood to quit index
- Linear inequality constraint involving hours worked by Bill and John
- Maximum total combined dollar cost per hour
- Maximum total combined work quality rating
- Maximum total combined likelihood to quit index

Given the constraints, we will define 'hours worked by Bill' as an integer variable since it cannot be a fraction, and 'hours worked by John' can be a continuous variable.

Here is how you could implement this in Gurobi:

```python
from gurobipy import *

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

# Define variables
Bill_hours = m.addVar(vtype=GRB.INTEGER, name="hours_worked_by_Bill")
John_hours = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_John")

# Objective function: Maximize
m.setObjective(5.02 * Bill_hours**2 + 4.37 * Bill_hours * John_hours + 3.47 * John_hours**2 + 3.78 * John_hours, GRB.MAXIMIZE)

# Constraints
m.addConstr(Bill_hours * 8 + John_hours * 13 >= 35) # Minimum total combined dollar cost per hour from hours worked by both squared
m.addConstr(Bill_hours * 2 + John_hours * 9 >= 41) # Minimum total combined work quality rating
m.addConstr(Bill_hours * 6 + John_hours * 12 >= 38) # Minimum total combined likelihood to quit index
m.addConstr(3 * Bill_hours - 6 * John_hours >= 0) # Linear inequality constraint

m.addConstr(Bill_hours**2 * 8 + John_hours**2 * 13 <= 123) # Maximum total combined dollar cost per hour from hours worked by both squared
m.addConstr(Bill_hours * 8 + John_hours * 13 <= 123) # Maximum total combined dollar cost per hour
m.addConstr(Bill_hours**2 * 2 + John_hours**2 * 9 <= 80) # Maximum total combined work quality rating from hours worked by both squared
m.addConstr(Bill_hours * 2 + John_hours * 9 <= 80) # Maximum total combined work quality rating
m.addConstr(Bill_hours * 6 + John_hours * 12 <= 71) # Maximum total combined likelihood to quit index

# Solve the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Bill: {Bill_hours.X}")
    print(f"Hours worked by John: {John_hours.X}")
else:
    print("No optimal solution found")
```