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.
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.
For the purposes of this, we will say the cable has Red, Green, Yellow, and Black conductors. They may be different colored or contain more conductors, but this is all we are concerned with. If the cable is solid with a matching stripe (Blue with a white stripe and white with a blue stripe twisted together, even better).
Connect both ends of the cable the same to an RJ45 JACK (do not crimp ends on the cable), essentially we are going to substitute color pairs of the existing cable for "standard" color codes in EIA/TIA 568B color code.
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).
Now substitute... If you cable is Red/Green Yellow/Black, connect the green to Pin 1, the Red to pin 2, the black to pin 3 and the yellow to pin 6, do the same on both ends and connect with a standard patch cable to the router and computer. If the color are different, just substitute as necessary.
Now, the reality this might work perfectly at 100Mbps, or it may error out like crazy... it may function fine at 10Mbps or fail just as miserably... Or it may not link up at all. There is no way of knowing until you try it.