You can right click on the project where you want to install newtonsoft nuget package.
You will find an option "Manage NuGet packages".
Then search online for "newtonsoft".
This would add a packages.config file in your project (depending on which framework you are targeting to ).
this new file contains all the nuget dependencies required for this project.
Refer this blog for more details
Answer from Manoj Choudhari on Stack OverflowHow can I download Newtonsoft directly/manually?
c# - How do I install Newtonsoft.json for Visual Studio 2010? - Stack Overflow
Update com.unity.nuget.newtonsoft-json to Newtonsoft.Json 13.0.2
Newtonsoft Json package - Unity Engine - Unity Discussions
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I mean without nuget.
How do I do that?
Btw, it's for Godot in case that matters.
You can right click on the project where you want to install newtonsoft nuget package.
You will find an option "Manage NuGet packages".
Then search online for "newtonsoft".
This would add a packages.config file in your project (depending on which framework you are targeting to ).
this new file contains all the nuget dependencies required for this project.
Refer this blog for more details
VS2010 is old and is unable to update to the latest NuGet version. This means that adding recent packages will often fail; however, you can often add an older version of a package that has a compatible NuGet version and it will work. This causes the suggested answer by @ManojChoudhari to fail for me. In the case of Newtonsoft.Json, the following worked using the Package Manager Console:
Install-Package Newtonsoft.Json -Version 9.0.1
I might be just be late to answer, but this is usually caused by the incorrect package source, different packages come from different sources, for example angularJS.core packages comes from nuget.org source so if you search it and you have Microsoft and .NET set as your source, then you wont be able to find it.
To attempt to answer your question, please select the change the source on your your PM console(see attached screenshot) 
, you might also want to make sure that you point to the correct project(attached screenshot will also guide you on that regard). Or even better just select "All" on your source and you should be good
What if you try to install it from Package manager UI. Right click in your solution/project and select manage nuget package. In the presented UI search for newtonsoft.json and install accordingly.
Also make sure your project don't already have it installed. In your project find the file name package.config and search for an entry named json.net. If present, delete that line, save the file and then try re-installing again.