Bonus

"No taxes will be deducted"

Yes, they will. This is Required. Otherwise, you are violating the law.

"and this will not be a separate payroll."

The Pay Date controls this factually. If this is the only activity on that date, it is its own Separate Payroll. If you add Bonus to regular paychecks or at least pay on the same Pay Date, then it is part of the One Payroll.

"Since we don't run payroll using QuickBooks, what is the best way to post an employee bonus check?"

You have two Options:

The same as every Other Paycheck process. You just happen to be using some Flat amount, not based on hours worked or some other measure.

Or, you give it out as Loan. Later, you need to run it through payroll as Gross, so that taxes compute and then you Deduct the Loan Amounts, to avoid paying it out again.

Example: The boss wants to hand out cash at the party tonight. Then, next week, everyone's paycheck includes the Bonus gross, taxes compute, the Cash handed out previously (tracked as Other Current Asset = Lending) is deducted, and their takehome this time is slightly lower than typical Hourly Paycheck, because it covered the taxes on the Bonus. But They Got More Money, as cash, at the party.