Does this product work on a Lenovo H530?

Yes it would, but from the lowest possible model of the H530 it is a very small graphics upgrade, and from every other model it is a graphics downgrade.
A Geforce 610 is a replacement for Intel integrated graphics between HD 3000 and HD 4000.
There might be a model with HD 2500, but a card like that is still a waste of money.
http://www.lenovo.com/...asheet.pdf

It has 2GB DDR3 ram. If you only have 4GB of system ram, you'll get just as much by adding in another 4GB stick.

And that card costs $48.
The H530 accepts full ATX height cards.
$50 gets a GT 720 at least
http://pcpartpicker.com/part/msi-video-card-n7202gd3lp
R7 240?
http://www.game-debate.com/...rce-gt-610
$53 after rebate
http://www.newegg.com/...6814150677

Not sure what graphics card you have. You would update or add drivers(software).
The power supply of the H530 is a piece of junk, as is every PC brand's PSU.
But, you can get as high as GTX 750 or R7 250 generally on the existing supply.
Benchmark list
http://www.videocardbenchmark.net/gpu_list.php

Yes, it works. That doesn't mean it is a good card to buy with money. It may be a graphics downgrade from your current level.
Install Belarc Adviser. Shut down and unplug your PC.
State your power supply and system-model number exactly.