Can a primary school teacher with a B.Ed. in one subject teach another subject?

For teaching to primary and upper primary (middle school) classes the minimum eligibility required for teachers is either A or B, as follows -
(A):
(i) Senior Secondary School Certificate or Intermediate.
(ii) Diploma or Certificate in basic teachers' training of not less than two years or Bachelor of Elementary Education (B.EI.Ed.).
OR
(B):
(i) Graduate with Bachelor of Education (B.Ed.) or its equivalent.

There is no mention of any specialization subject in this eligibility and it is obvious that for primary levels that is not required as the teacher is supposed to teach all the subjects.
This framework is mainly for the state Govt and central Govt schools and I am not sure whether private schools are following it or not.
I also assume that schools recognised by the Govt should be following the same pattern.

In Primary schools, teachers are not appointed subject-wise. A teacher should be able to teach all the subjects of a particular class that was assigned to him.
Actually, graduates with B.Ed qualifications are supposed to be appointed as B.Ed assistants and they have to teach a subject to all classes. These teachers will be appointed in high schools and they have to teach students of different classes starting from the 6th Class to the 12th class.
The eligibility criteria for a teacher to become a secondary-grade teacher are
1. A pass in 10th class/SSC or 12th Class/ Intermediate.
2. Secondary grade training certificate or any other equivalent training certificate recognised by the concerned authorities.
But as there is a lot of demand for these jobs even candidates with graduation and a B.Ed can also apply for these posts. But they should teach all the subjects.
I feel it is not at all a difficult task to teach all the subjects to a primary class student for a B.Ed if really he has a flair for teaching.

If the teaching assignments is to be done in a primary school, the aspirant should be capable enough to teach all the subjects of all the classes included in the syllabus. Though B.Ed is often pursued having passed the graduation irrespective of the streams and are assigned the job of teaching in a high/ higher secondary school but they should have the potentials of teaching of all the subjects of primary levels.
The teachers having acquired a B.Ed degree followed by graduation are appointed in the high schools to teach the pupils right from class six to class twelve expecting them to be involved in teaching the different subjects of the pupils of all the classes with ease.
The eligibility criteria to teach pupils of class six to eight is different as indicated below-
1) A pass of class ten/ Intermediate.
2) They have to pass secondary school certification recognised by the concerned authorities.
However, sometimes the school management would like to recruit teachers of B.Ed degree teachers expecting them to take up the teaching assignments of all the subjects irrespective of the classes in the school by virtue of their qualifications and hence they recruit teachers having B.Ed qualifications.

I am teaching in a primary school in Madhya pradesh. And yes it is valid to teach all subjects like I am teaching English, Science and Social science from Class 4th to Class 7th.

As long as you feel comfortable, you can pursue your present assignment. There should not be any problem from your school authorities. On the strength of your qualification being B.Ed, your school administration is very sure of your competence in the subjects which you are teaching. Your qualification differentiates you from others.
If you have any inclination for taking up teaching in other subjects such as Hindi, Mathematics, Sanskrit etc being taught in primary level, talk to your Headmaster/ Secretary. In that way, you will get immense satisfaction where as your school authority would identify your extra potentials. This would further boost your career prospects.