It is illegal to pay an H1-B Visa Holder differently to the prevailing wage. Converting to a Green Card should have no effect on your pay, and if it does, the employers are committing fraud.

You may have a side-effect of increased pay due to having more positions open to you. Holding an H1-B restricts you to a specific job description. For example, if you were a plain ordinary "Software Engineer" with nobody reporting to you, then you are not normally allowed to move to a role where you would have reports. This can be overcome with due-process and lawyers etc, but the basic idea is an H1-B worker is coming to the US to fill a job role not just a job.

So, when you convert to holding a Green Card, you can get promoted into positions which weren't previously open to you. You get the pay increase by being a better engineer and getting the better job.