what is the size or model of an add-a-tee(similar stuff) can convert a toilet water outlet working both for a toilet water tank and a washing machine
Assuming this is a rental, I do not recommend this at all. If anything goes wrong, and things can go very wrong, you will have a flood on your hands and your landlord will not be happy.
If you own this apartment, I do not recommend this at all. The chances of things going wrong are relatively big, and even if everything works correctly, the flow rate will be very low because this connection is designed for a toilet and won't simply &quot;scale up&quot; to fill a washing machine.
Think about all the pieces needed, normally, for a washing machine:

Cold water in
Hot water in
Electricity
Waste water out

This is specifically for cold water in.
Does your washing machine need hot water as well? Or does it heat the water if a cycle calls for hot water?
Electricity is probably the easiest piece, at least for a washing machine. Most US/Canada washing machines use a standard 120V 15A receptacle. On the other hand, if you want to install a dryer then you typically need 240V and also figure out how to handle venting the hot air out.
What about waste water drainage? That may be more than you realize. A typical washing machine pushes out the water very quickly, possibly faster than your sink drain can take it.
What I suggest looking at is a &quot;portable&quot; washing machine. These typically get water using an adapter on a regular kitchen faucet. No permanent installation or changes to existing plumbing, which solves the landlord problem (the only effect on the landlord is higher water usage) and solves the &quot;things can go wrong&quot; problem, at least with respect to plumbing installation.