Can I run "Gigabyte GeForce GTX 1050 Ti" gpu in GA-H61M-S2PV (rev. 2.2) motherboard ?

Solution
Yes you can use that graphics card with that build and if you can upgrade CPU to a core i5 (even a used one) would make it even better.

I think the Core i3 2120 cannot manage to run the PCIE slot at gen 3 sepcs but it's just fine if it works at PCIE gen 2 specs.

What is your PSU model and capacity? At least a 300W PSU is needed and some 3rd party makes/models require a 6-pin power cable from the PSU.
Yes you can use that graphics card with that build and if you can upgrade CPU to a core i5 (even a used one) would make it even better.

I think the Core i3 2120 cannot manage to run the PCIE slot at gen 3 sepcs but it's just fine if it works at PCIE gen 2 specs.

What is your PSU model and capacity? At least a 300W PSU is needed and some 3rd party makes/models require a 6-pin power cable from the PSU.
 
Solution
What brand/model is the PSU? Is it modular? Most cables are detachable from it or a bunch of them sticking out a corner of the unit? Asking because you said you don't have a 6-pin cable.

According to the Gigabyte website that GPU requires a 6-pin power cable from the PSU.

https://www.gigabyte.com/Graphics-Card/GV-N105TG1-GAMING-4GD#sp

https://static.gigabyte.com/Product/3/6054/20161014181609_src.png


If you're PSU does not have that or (I'm guessing) it's a few years old and you can get a good reliable 400W unit it would work fine.