Yes, but either the Ethernet cables connecting the two devices would need to plugged into a LTE-existing network that will assign IP addresses to the devices in question, or you would have to somehow manually assign static IP addresses to each one (how depends on the controlling device) and then use those addresses when attempting to establish the connection.
“Serial to Ethernet” cables also aren’t terribly common, and so I’m wondering whether the device truly implements Ethernet on the RJ45 side or whether it actually uses a proprietary protocol that requires a matching receiver at the other end and the fact that it uses a commonly available twisted pair cable is just a coincidence. After all, “serial” (presumably RS-232 or a similar asynchronous behavior) is essentially raw bits and nearly protocol-less, whereas a full “Ethernet” implementation requires some processing horsepower and a means of addressing and formatting the data into packets, and it’s not clear how you’d get from one side of this device to the other. At minimum there’d need to be some means of controlling the behavior of the Ethernet side or setting the baud rate and start/length/stop/parity parameters of the serial side, for example.
