Hi suraj,
You can use a Payment Gateway like Paytm or any other and can integrate it using REST API by making callouts.
The Payment Gateway to which you want to integrate it with , reach out their website for their exposed urls which will be required while making API Calls.And remember you cannot make callouts just after DML’s and sometimes while making callouts in Contructor of any Controller , it gets timeout notification.For more about How to do integration using REST API , refer to this article.
Thanks.