If you are trying to change the page layout for pre-created SharePoint default page than you will not get it (If you are changing the page layout of the page which are created before activating feature than it will not work.).
The issue is that the feature is not enabled that allows the layouts.
To fix the issue go to Site Actions -> Site Settings in the upper right corner.
Under "Site Collection Administration" click on "Site Collection Features".
Look for "SharePoint Server Publishing Infrastructure" and activate it. It might take a moment to load.
Next return to "Site Settings" and click on "Manage Site Features"
Look for "SharePoint Server Publishing" and activate it. It might take a moment to load.
If you don't have the Site Collection Administration section of your settings you can try skipping to the next part. However if the site collection does not have this enabled then you will get an error when trying to activate it on the site. You will need to get into the site collection's options.
After making a new site you can go into Site Features and again enable SharePoint Server Publishing. This worked fine on a blank webpage so a publishing page/site is not necessary.