Can I put an Ethernet port in my brick wall?
Two issues - cable and port.
Cable through brick wall
If one end is on one side of a brick wall and the other end is on the other side of the brick wall, drill a hole with a hammer drill and a masonry bit and fish the wire through. A brick wall could be just a couple of layers of brick but often has cinderblock or (as I found when I replaced my oven years ago) all kinds of stone, concrete and junk inside. Ideally you want a bit long enough to go all the way through everything in one shot. Not an easy job, but usually not too hard with a decent drill and bit.
Cable on surface of brick wall
If you can't easily go through the wall, or you need to run the cable a distance along the wall for other reasons (e.g., the router is 20 feet down the wall on the same side) then use a surface raceway. Basically, a plastic or metal rectangular tube that you mount on the wall and stick the wire inside. Some people don't like the look, but running cable inside a brick wall is not practical so it is either on the surface in a raceway (better than just leaving the wire loose) or through to the other side. If you have a baseboard or molding on the bottom of the wall then you can, depending on the details, either mount a raceway to the baseboard/molding easier than mounting to the brick, or staple the cable directly to the baseboard/molding.
Depending on the type of wall (interior vs. exterior) and where you need to run the cable, and what is underneath the floor, you may be able to drill down through the floor instead and run the cable through the room or crawlspace underneath. However, that is sometimes tricky too depending on the ceiling of the lower level.
Port
Surface mount the port. Do NOT try to cut a hole for a box into the brick - it can be done but is absolutely not worth the effort. You can either use a standard box designed for mounting on the wall and put a regular Ethernet wall plate on the box, or use a smaller surface mount port. A standard box has the advantage of allowing for 2 or more ports - easily up to 6, for additional Ethernet or phone or cable TV.