Hello @Dua_Fatima,
Welcome to the forums! 
Yes, you can calculate alpha diversity stats on a per-sample basis. This is because alpha diversity is a per-sample measure (like number of different features observed in a sample), while beta diversity compares two samples (how many more features are in this sample compared to that one).
Stephanie put together this awesome list of all the alpha and beta diversity stats, which you might find helpful. That explains the alpha metrics you were able to calculat, like Faith's PD and Chao 1.
Most of the tutorials and examples use many samples as that's more common, and more informative. Do you have a larger cohort of samples you will be looking into in the future?