How can I use coaxial cable for ethernet

Solution 1:

You bought the wrong thing.
These converters convert analog video-signal (on coax) to UTP (unshielded twisted pair) so you can forward the analog signal from a CCTV camera via existing UTP network cable to somewhere else and there convert it back to analog video to display on a monitor or feed it to a VHS.
camera <-> coax <-> UTP <-> coax <-> monitor
You can't use them in reverse to do: PC <-> UTP <-> coax <-> UTP <-> router.

Ether does run over coax. Actually it started there before we had UTP cables.
But for that you needed ethernet cards with a coax plug (and some other stuff as well, it isn't a plain point-to-point connection). And anyway it used a different type of coax than television does. Television coax won't work with these old network cards.
These things went the way of the dodo around the year 2000 and good riddance too. It was very slow compared to modern network technology.

What you really need is a MoCA (Multimedia over Coax Alliance) converter.
See this page for certified equipment and vendors: http://www.mocalliance.org/products/index.htm

Be advised that MoCA stuff is probably more expensive then pulling a CAT5E or CAT6 cable to the garage. (If the existing coax is in a pipe you may be able to attach an UTP cable to one end and use the coax to pull the UTP through the pipe.)

Solution 2:

Presumably you have power in the garage (or why have network connectivity out there) so maybe powerline adaptors might be worth looking at. It won't use the coax but is that mandatory?

Solution 3:

A better answer may be to investigate using a WiFi router in your office and just use WiFi to connect up within your house. 30 feet really isn't very far for modern WiFi routers (especially if you get one that's high power) even through walls and structure. May be a more turn-key solution than trying to route Ethernet across your coax cable.