I work for a property management company, and we have a trust account that all the rents are paid into, and then we write checks out for the net rent to the property owners. My question is, if we need to pay a bill for a property owner before they pay us for it, can we write it out of the trust account or do we need to have that specific payment in hand before we can pay it?

There are two places you need to look for an answer to your question. First, see whether your property management contract states that you may (or must, perhaps) advance funds on behalf of the owners. Second, see if your states laws address the responsibilities of property managers in your position.