There are several ways to count the objects in S3 bucket. But, there is deficiency regarding the API calls which has its own price. So here are some ways that I usually do to count the objects.

1. By using aws cli

I know this is the programmatically way to get the total objects, but beware for the GET api calls pricing, if there's too much files it will cost you a lot of money.

  aws s3 ls s3://<bucket_name>/ --recursive | wc -l

2. By using web console

Go to buckets > bucket-name > Metrics.

You will see some metrics like total bucket size, total number of objects, and more.