Does an employer have to pay a terminated employee bonus if they where terminated the first day of a new quarter?

It depends on what the policy or contract stipulates. Talk with your legal counsel.

Agree with Wayne, the policy definitions control.

In any event if you are forcing the separation you should err on the side of paying the bonus. This will go a long way to keeping the other potential issues that could arise at a low boil. How much is it really going to cost you to show some goodwill to avoid them wasting your time taking you to court? Cost benefit says pay them their bonus if you have failed to properly define your policy.

I would concur with plan terms control the subsequent administering of the payouts. Keep in mind, if the bonus is "discretionary" in nature (e.g. not tied to pre-determined individual performance targets), there is no "guarantee" of a payout. If it is clearly designed as an incentive, you many need to pro-rate it. Regardless of being employed during the performance period, many companies will include the clause "must be active on payroll" to receive payout. At the end of the day, you want to comply with the terms of the plan and not make exceptions that can raise issues of bias or "discrimination". Hope this helps.

Concur with Malak, here. Check the bonus plan, and state rules, but the "must be employed through the end of the bonus period" is an enforceable clause, and needs to be enforced consistently.