It's difficult to say. In Oregon, restraining orders (sometimes called FAPA orders) are civil matters, not criminal. If the background check was only for criminal matters, it presumably would not come up. However, a civil restraining order is still part of the public court record. A background check which looks for court actions in general, not just criminal matters, would likely reveal a restraining order. Generally, you should assume that if it's part of the public record, it can be discovered.