Physics says no. Why? Twisted pair Ethernet has those twists for a specific reason so while you could wire what you have to Ethernet jacks it isn't the right cable for the job due to the lack of twists and pairs.
I'm not saying you can't electrically wire it up, I'm writing it would have issues at Ethernet speeds.
There is no need for any adapter or convertor. Just RJ45 ends and the crimper. And there are now no-crimp RJ45 ends. Example: https://www.google.com/search?q=no+crimp+rj45
Let's cover the other way to connect when there is no cable. PowerLine Ethernet. I always liked this one as it does it all. Picture follows:
Hi, I have a 5 pair * 0.5mm telephone cable which run through concealed wiring between my master bedroom and my bedroom. 2 wires out of 5 pairs are used for telephone connection so I crimped RJ45 connector to rest of 4 pairs on both sides so that I can connect one end of it in my bedroom router ethernet lan port and second end to the computer in master bedroom. I have a wifi router in my bedroom but it's signal is very week in master bedroom, I had changed a couple or routers but still the same signal strength so i thought the wired solution would be best. I had asked the electrician if he can replace the old 5 pair telephone cable concealed in wall with a ethernet cable, to which he said it will take lot of effort since the cable was concealed long back in the wall and due to multiple cables running through a small concealed wiring pipe it's going to be very difficult to replace the cable and if i still want it then he will have to run a new cable through new concealed pipe which would be time consuming and very expensive too.
So I thought I could simply use the existing 5 pair telephone wire as ethernet cable but after crimping the RJ45 connector and connecting one end to the lan ethernet port on my bedroom router and another end to the computer lan port in the master bedroom, I tried to check the lan connection on the master bedroom computer it doesn't work and it shows no lan connection.
Now in order to check if the wire is properly crimped and there is no breakage in the wire in the wall, i thought of testing it through a different way .i.e. I used a RJ45 to VGA adapter and connected the vga port of my bedroom computer CPU to the monitor in the master bedroom and guess what I was able to view the content played on my bedroom cpu on the master bedroom monitor, this means the cable is working fine to carry a video signal from my bedroom computer to master bedroom computer monitor but somehow it is unable to send the internet signal from one end to another end. I believe i can use some converter on both ends which could resolve this issue. Can you please advise if there is any converter in the market which could help resolve this problem.