Yes, if you're married to a U.S. citizen, you can apply for a green card (adjustment of status) from inside the U.S. even if your H-1B is expiring soon.

The process typically starts with your spouse filing Form I-130 and you filing Form I-485 (green card application) at the same time (a "concurrent filing").

 If your H-1B is close to expiring, it’s best to file your green card application as soon as possible to avoid gaps in your status.

If your H-1B expires while your adjustment of status is pending, you can still remain in the U.S. lawfully while your green card is processed.