The answer is possibly... but without knowing more about the cable and the layout it is difficult to tell.

In theory, telephony cable can be used to support 10/100 connections (not Gigabit) over short distances. If you can find a telephony cable that is not cut and is a "home run" between the two points you need to connect, and it contains at least 4 conductors, it may be possible. RJ22, also called an RJ10, is the smallest of the modular plugs. It has a 4P4C configuration and is used in handset cords. It is not an official ACTA (Administrative Council for Terminal Attachments) specification. RJ11 is used for connecting single phone lines and uses a 6P2C or 6P4C configuration. With only 2 of the 6 available connection points, the RJ11 plug supports limited-bandwidth applications. This includes phone lines and ADSL connections.

For 10/100 Ethernet to work, you need to have a straight through connection of pins 1, 2, 3, and 6, and the other pins are not relevant/not used in this scenario. RJ12 connectors are also similar to RJ11 but have a 6P6C configuration. They are used to connect a phone line with the key telephone system, small exchange-like systems used by businesses to connect to the carrier on the outside and create a network of phones and extensions on the inside. RJ12 connectors can be used in place of RJ11 because RJ12 has all the connectors RJ11 needs (plus 4 extra).

In the 568B color code, we need W/O-O/W (Pins 1&2) and W/G-G/W pairs (Pins 3&6) Pins 4, 5, 7, and 8 can be ignored (they are for PoE and Gigabit applications, which we are not going attempt with this type of cable).