Hey, if suppose TRV is rejected, and we will apply for conjugal. Does IRCC will ask us why we didn't married in other countries and why I can't go back to India for one year to establish a common law relationship? How I could explain?
So I don't have a complete answer, you'll have to go through the application package and likely other threads here.
-For common law: I think the simple answer would normally be that you have employment and commitments in Canada and as a PR wish to comply with your residency obligation. (Of course you still might conisder that option if it works for you).
-I don't know as well about getting married in other countries but one normal plausible answer would be what countries your partner could travel to (i.e. has a visa) that will perform same sex marriages for foreigners. (I don't
think that there is a requirement to travel long distances to do this - but if the partner travels frequently to USA, you can expect the question would come up with not many good answers).
That said - if marriage in a third country is an option, it might be preferable for you. (For probably obvious reasons, conjugal sponsorship is subject to a fair amount of scrutiny - i.e. since the partners have not married nor become common law, they will need to be convinced it is a genuine relationship, which can be hard to demonstrate). For example if your partner had a visa to countries like usa / much of Europe, that would be easier.
My impression is that for same sex couples, the common problem is that most regional countries/visa-free for easier travel do not allow same sex marriage.
Didn't realise you're from India, this is further complicated right now by travel bans (and possibly limited TRV processing).
Still: your best starting point is still ... you'll have to and want to apply for the TRV anyway.