Can we apply for green cards to become permanent residents since our children were born here last year and we're on H1-B?

Not now... your child will be eligible to sponsor immigrant visa for parents when he/she is 21 years old, has sufficient income, pay taxes, etc.

Thanks lawanwadee for your prompt answer. Do you have any suggestions how to live permanently in the US since the H1B is always dependent on the employer and for six years max.

Get green card by other means:

1. through job offer (apply for perm LCA) and submit I-140. When approved you will keep your H status until you may apply for green card.
2. through refugee status - if it is dangerous to return to your home country seek refugee status after your h status is over
3. through US army - you will get instant citizenship if go to war zone, or in 6 month if duty is in US. You will have to complete with honor to approve it.
4. every year participate in green card lottery.
5. there may be other possibilites as well (become famous, get different education and get a job, that you can use for instant green card application, etc.)