Completed MSc in 2013 but did not collect migration and conduct certificates

You have to request them for these records. If they say that these are old records and they do not have a proof that you have not collected the same earlier then they might consider issuing the duplicate of the same to you. What I would suggest is please send a request letter or mail giving all your details and circumstances in which you could not collect these certificates after completing your course from there. If you are studying further then migration certificate becomes important. Character certificate you can get from your recent employer also and that is not a big deal. If it is possible for you to visit that institution and meet the principal and request him for this favour then that would be the best thing to do in this situation. Nine years is a long time and they might raise many excuses like the record missing or say that they are not able to locate the particular record and things like that.
As I understand every institution has a process of delivering records on payment of a fee and you also indicate that you are ready to pay the fee for that service. personal meeting would be very useful in such matters. If they tell you that they are not sure of your verifications and are not sure to issue the certificates then you can suggest them that you would bring some affidavit duly attested by notary public and things like that and after getting those legal procedures they might consider your case as the genuine one. So try all these methods and I hope that you will be able to get those old records dug there. All the best.

The universities and colleges are supposed to keep their record available for many years.
I lost of 10th certificate after almost 20 years after passing the same. That is the only date of birth proof I have. So I applied for a duplicate copy and I got the same 45 days after applying. All universities will have some way to keep old records.

A migration certificate will be issued when you apply only. Otherwise, by default, you will not get that migration certificate. Only when we apply for it only they will give it. This is will be given to us when we apply for it if you are going for further studies at another university.
So you can apply to the university in which you studied for your M.Sc with all the required details as per the procedure. You can see the procedure online on the website of the university and you will get your migration certificate.

Coming to the conduct certificate you have to get it from the college where you did your M.Sc. You can meet the principal of the college and explain to him the reason for your asking for the same. They will issue the same.

You would accept that you have made an inordinate delay in collecting your migration and conduct certificate though you have passed your M.Sc way back in 2013.
Migration certificate is the essential certificate especially when you want to continue your studies further in other university or any academic institution, you need to procure the said certificate so that you could continue your higher studies. Character Certificate would not create any problem since the same can be procured from the institute where you are working currently.
Now the following steps are to be taken from your end -
1) Since a substantial delay has occurred from your end in the collection of your migration certificate, it would be better on your part to visit personally to the university office and meet the appropriate authority apprising him of the facts regarding your late initiatives. If the authorities ask you for a written statement, oblige him for your speedy process.
2) In your application, provide the details of year of passing out M.Sc.( in your case 2013) and Certificate No, Your Registration No etc so as to connect to your migration certificate with the existing No of your post graduation certificate.
3) You need to meet some clerk/ officer taking care of past records and try to approach the same officer. In that way, you could locate your migration certificate.
4) You need to submit the prescribed fees fixed for the collection certificate and other service charges if any in the university counter and procure the receipt.
Hope you could manage your situation with ease with the observance of these steps.