Contact template doesnt appear in page template options

Page templates require only a few things to work:

1. To be in the theme’s (or child theme’s) directory. Either in the root directory or in a subfolder.
2. To have appropriate permissions (It will need to be read by WordPress, usually www user). Typically it should have the permission 644 or 640.
3. To have a Template name header

From the question and comments you have (1) and (3), so the file you have created presumably has the wrong file permissions.