Not with the power injectors you have. Ubiquiti has 2 types. They use 24 volt passive power injectors, these like the cisco ones are proprietary.
The other kind are the standard 802.3af/at models. These you can hook to any standard PoE switch without a special power injector. They of course sell power injectors that run 802.3 standards if you want separate units.
You have to look through their line of device they actually have some that can both their proprietary and 802.3 based ones...not sure how they did that. It very clearly tells in the product description how it is powered.
It is pretty common for any device that is called a AP to be powered via ethernet somehow. That really is the only main difference between a actual AP and running a router in AP mode.