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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<T> interface. So you better just do it like

public class UserDtoMapper : IMapper<UserDto>
{
 public UserDto Map(SqlMapper.GridReader reader)
 {
 return reader.Read().Select(x => 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<T>
{
 private IMapper<T> mapper; 
 public QueryObject(IMapper<T> mapper)
 {
 this.mapper = mapper;
 }
 public T Query(int id)
 {
 ..........
 return mapper.Map(reader)
 ...
 }
} 

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<T> interface. So you better just do it like

public class UserDtoMapper : IMapper<UserDto>
{
 public UserDto Map(SqlMapper.GridReader reader)
 {
 return reader.Read().Select(x => 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<T>
{
 private IMapper<T> mapper; 
 public QueryObject(IMapper<T> mapper)
 {
 this.mapper = mapper;
 }
 public T Query(int id)
 {
 ..........
 return mapper.Map(reader)
 ...
 }
} 

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<T> interface. So you better just do it like

public class UserDtoMapper : IMapper<UserDto>
{
 public UserDto Map(SqlMapper.GridReader reader)
 {
 return reader.Read().Select(x => 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<T>
{
 private IMapper<T> mapper; 
 public QueryObject(IMapper<T> mapper)
 {
 this.mapper = mapper;
 }
 public TQuery(int id)
 {
 ..........
 return mapper.Map(reader)
 ...
 }
} 

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<T> interface. So you better just do it like

public class UserDtoMapper : IMapper<UserDto>
{
 public UserDto Map(SqlMapper.GridReader reader)
 {
 return reader.Read().Select(x => 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<T>
{
 private IMapper<T> mapper; 
 public QueryObject(IMapper<T> mapper)
 {
 this.mapper = mapper;
 }
 public T Query(int id)
 {
 ..........
 return mapper.Map(reader)
 ...
 }
}