Demo.cs

using Newtonsoft.Json;
using NUnit.Framework;

public class Demo
{
    private JsonSerializerSettings _settings;
    [SetUp]
    public void SetUp()
    {
        _settings = new JsonSerializerSettings()
        {
            ContractResolver = new SnakeCaseContractResolver()
        };
    }

    [Test]
    public void Deserilaize()
    {
        var json = "{\"first_name\":\"Johan\"}";
        var dto = JsonConvert.DeserializeObject<Dto>(json, _settings);
    }

    [Test]
    public void Serialize()
    {
        var dto = new Dto("Johan");
        var json = JsonConvert.SerializeObject(dto, _settings);
        Assert.AreEqual("{\"first_name\":\"Johan\"}", json);
    }
}

Dto.cs

public class Dto
{
    private Dto()
    {
    }
    public Dto(string firstName)
    {
        FirstName = firstName;
    }
    public string FirstName { get; private set; }
}

SnakeCaseContractResolver.cs

using System;
using System.Linq;
using System.Reflection;
using System.Text.RegularExpressions;
using Newtonsoft.Json;
using Newtonsoft.Json.Serialization;

public class SnakeCaseContractResolver : DefaultContractResolver
{
    private readonly string[] _prefixable = { "name" };
    protected override JsonProperty CreateProperty(MemberInfo member, MemberSerialization memberSerialization)
    {
        JsonProperty prop = base.CreateProperty(member, memberSerialization);

        string propertyName = prop.PropertyName;
        if (_prefixable.Any(x => this.TryPrefix(propertyName, x, out propertyName)))
        {
            prop.PropertyName = propertyName;
        }
        prop.PropertyName = propertyName.ToLower();
        return prop;
    }

    private bool TryPrefix(string input, string prefix, out string prefixed)
    {
        if (input.IndexOf(prefix, StringComparison.OrdinalIgnoreCase) > 0)
        {
            prefixed = Regex.Replace(input, prefix, string.Concat("_", prefix.ToLower()), RegexOptions.IgnoreCase);
            return true;
        }
        prefixed = input;
        return false;
    }
}