Yes, you can can keep bees in Gresham. It's a county thing. Officially you're suppose to get a permit from the county. It's actually pretty easy, they just want your money.
There is an application somewhere online that you fill out. All it involves is making sure anyone within 150 feet from your hives is okay with it. You get them to sign, and then send it in.
When I did mine the neighbors were all very cool with it. Invite them over to do a hive inspection and make sure to give them honey at Christmas. It's a great chance to educate them. Most of my neighbors don't use toxic chemicals in their yard anymore because they know if the bees do well, they'll get another jar of honey. It's also a great way to recruit help when it comes to extracting!
The permit is actually fairly cheap, $10 when I got mine a few years ago.