Querying a database with Dapper
Setting aside that you won't go with a Mapping framework like Jeroen Vannevel suggested in his comment, this basically looks good but can still be improved.

I don't see any value added by creating an abstract class which implements the IMapper&lt;T&gt; interface. So you better just do it like  

public class UserDtoMapper : IMapper&lt;UserDto&gt;
{
    public UserDto Map(SqlMapper.GridReader reader)
    {
        return reader.Read().Select(x =&gt; new UserDto()
        {
            Id = x.Id,
            Name = x.Name
        }).SingleOrDefault();
    }
}


But if you travel this route you should do it right and also make QueryObject class generic like  

public class QueryObject&lt;T&gt;
{
    private IMapper&lt;T&gt; mapper; 

    public QueryObject(IMapper&lt;T&gt; mapper)
    {
        this.mapper = mapper;
    }

    public T Query(int id)
    {
        ..........
         return mapper.Map(reader)
        ...
    }
}