NEST do not operate a 'Net Pay Arrangement' so the employee's contribution is paid after deduction of tax, and taken from their 'net (taxed) pay'
So the contribution is calculated as:-
Gross Pay x 5 % x (!- Basic rate of tax(20%)=4% of employees gross salary
So for a gross salary of £719 an employee payment of £28.76 would be correct
The 'missing 1% is reclaimed from HMRC by NEST
In Quick books you need to ensure that all earnings are included in the calculation of pension contributions, and set the contribution rate to 4% (not 5%) - and then say that this does not affect any of the taxes.
I am not sure how QB could arrive a contribution of £8.30 but suspect you have not included all the relevant earning.
EDIT
Excluding earnings below the lower earnings threshold would result in contributions of 4% (after tax) on approx £200 pounds of income = approx £8.30 / month.
So the problem relates to the application of the lower earning threshold, which is not mandatory - but must be consistent in NEST and QuickBooks